\documentstyle[11pt,reduce]{article}
\title{GROEBNER: A Package for Calculating Gr\"obner Bases, Version 3.0}
\date{}
\author{
H. Melenk \& W. Neun \\[0.05in]
Konrad--Zuse--Zentrum \\
f\"ur Informationstechnik Berlin \\
Takustrasse 7 \\
D--14195 Berlin--Dahlem \\
Germany \\[0.05in]
Email:  melenk@zib.de \\[0.05in]
and \\[0.05in]
H.M. M\"oller \\[0.05in]
FB Mathematik \\
Vogelpothsweg 87\\
Universit\"at Dortmund \\
D--44221 Dortmund \\
Germany\\[0.05in]
Email: moeller@math.uni--dortmund.de}

\begin{document}
\maketitle

\index{Gr\"obner Bases}
Gr\"obner bases are a valuable tool for solving problems in
connection with multivariate polynomials, such as solving systems of
algebraic equations and analyzing polynomial ideals. For a definition
of Gr\"obner bases, a survey of possible applications and further
references, see~\cite{Buchberger:85}. Examples are given in \cite{Boege:86},
in \cite{Buchberger:88} and also in the test file for this package.

\index{Groebner package} \index{Buchberger's Algorithm}
The $groebner$ package calculates Gr\"obner bases using the
Buchberger algorithm.  It can be used over a variety of different
coefficient domains, and for different variable and term orderings.

The current version of the package uses parts of a previous
version, written by  R. Gebauer, A.C. Hearn, H. Kredel and H. M.
M\"oller. The algorithms implemented in the current version are
documented in \cite{Faugere:89}, \cite{Gebauer:88},
\cite{Kredel:88a} and \cite{Giovini:91}.
The operator $saturation$ has been implemented in July 2000 (Herbert Melenk).
\section{Background}

\subsection{Variables, Domains and Polynomials}

The various functions of the $groebner$ package manipulate
equations and/or polynomials; equations are internally
transformed into  polynomials by forming the difference of
left-hand side and right-hand side, if equations are given.

All manipulations take place in a ring of polynomials in some
variables $x1, \ldots , xn$ over a coefficient domain $d$:
\[ d [x1,\ldots , xn], \]
where $d$ is a field or at least a ring without zero divisors.
The set of variables $x1,\ldots ,xn$ can be given explicitly by the
user or it is extracted automatically from the
input expressions.

All \REDUCE \  kernels can play the role of ``variables'' in this context;
examples are

%{\small
\begin{verbatim}
x y z22 sin(alpha) cos(alpha) c(1,2,3) c(1,3,2) farina4711
\end{verbatim}
%}

The domain $d$ is the current \REDUCE \  domain with those kernels
adjoined that are not members of the list of variables. So the
elements of $d$ may be complicated polynomials themselves over
kernels not in the list of variables; if, however, the variables are
extracted automatically from the input expressions, $d$ is identical
with the current \REDUCE \  domain. It is useful to regard kernels not
being members of the list of variables as ``parameters'', e.g.
\[
\begin{array}{c}
 a * x + (a - b) * y**2 \;\mbox{ with ``variables''}\ \{x,y\} \\
\mbox{and ``parameters''  $\;a\;$ and $\;b\;$}\;.
\end{array}
\]

The exponents of $groebner$ variables must be positive integers.

A $groebner$ variable may not occur as a parameter (or part
of a parameter) of a coefficient function. This condition is
tested in the beginning of the $groebner$ calculation; if it is
violated, an error message occurs (with the variable name), and the
calculation is aborted. When the $groebner$ package is called by
$solve$, the test is switched off internally.

The current version of the Buchberger algorithm has two internal
modes, a field mode and a ring mode. In the starting phase the
algorithm analyzes the domain type; if it recognizes $d$ as being a
ring it uses the ring mode, otherwise the field mode is needed.
Normally field calculations occur only if all coefficients are numbers
and if the current \REDUCE \  domain is a field (e.g. rational numbers,
modular numbers modulo a prime). In general, the ring mode is faster.
When no specific
\REDUCE \  domain is selected, the ring mode is used, even if the input
formulas contain fractional coefficients: they are multiplied by their
common denominators so that they become integer polynomials. Zeroes of the
denominators are included in the result list.

\subsection{Term Ordering} \par
In the theory of Gr\"obner bases, the terms of polynomials are
considered as ordered. Several order modes are available in
the current package, including the basic modes:
\index{lex ! term order} \index{gradlex ! term order}
\index{revgradlex ! term order}

\begin{center}
$lex$, $gradlex$, $revgradlex$
\end{center}

All orderings are based on an ordering among the variables. For
each pair of variables $(a,b)$ an order relation must be defined, e.g.
``$ a\gg b $''. The greater sign $\gg$  does not represent a numerical
relation among the variables; it can be interpreted only in terms of
formula representation: ``$a$'' will be placed in front of ``$b$'' or
``$a$''  is more complicated than ``$b$''.

The sequence of variables constitutes this order base. So the notion
of
\[ \{x1,x2,x3\} \]

as a list of variables at the same time means
\[ x1 \gg x2 \gg x3 \]
with respect to the term order.

If terms (products of powers of variables) are compared with $lex$,
that term is chosen which has a greater variable or a higher degree
if the greatest variable is the first in both. With $gradlex$ the sum of
all exponents (the total degree) is compared first, and if that does
not lead to a decision, the $lex$ method is taken for the final decision.
The $revgradlex$ method also compares the total degree first, but
afterward it uses the $lex$ method in the reverse direction; this is the
method originally used by Buchberger.

\example \ with $\{x,y,z\}$: \index{Groebner package ! example}
\[
\begin{array}{rlll}
\multicolumn{2}{l}{\hspace*{-1cm}\mbox{\bf lex:}}\\
 x * y **3 & \gg & y ** 48 & \mbox{(heavier variable)} \\
 x**4 * y**2 & \gg  & x**3 * y**10 & \mbox{(higher degree in 1st
variable)} \vspace*{2mm} \\
\multicolumn{2}{l}{\hspace*{-1cm}\mbox{\bf gradlex:}} \\
  y**3 * z**4 & \gg & x**3 * y**3 & \mbox{(higher total degree)} \\
  x*z  &        \gg & y**2  & \mbox{(equal total degree)}
\vspace*{2mm}\\
\multicolumn{2}{l}{\hspace*{-1cm}\mbox{\bf
revgradlex:}} \\
 y**3 * z**4 & \gg &  x**3 * y**3 & \mbox{(higher total degree)} \\
 x*z         & \ll  &  y**2       & \mbox{(equal total degree,} \\
 & & & \mbox{so reverse order of lex)}
\end{array}
\]

The formal description of the term order modes is similar to
\cite{Kredel:88}; this description regards only the exponents of a term,
which are written as vectors of integers with $0$ for exponents of a
variable which does not occur:
\[
\begin{array}{l}
  (e) = (e1,\ldots , en) \;\mbox{ representing }\; x1**e1 \ x2**e2 \cdots
  xn**en. \\
  \deg(e) \; \mbox{ is the sum over all elements of } \;(e) \\
  (e) \gg (l) \Longleftrightarrow (e)-(l)\gg (0) = (0,\ldots ,0)
\end{array}
\]
\[
\begin{array}{rll}
\multicolumn{1}{l}{\hspace*{-.5cm}\mbox{\bf lex:}} \\
  (e) > lex > (0) & \Longrightarrow  & e_k > 0 \mbox{ and } e_j =0
\mbox{ for }\; j=1,\ldots , k-1\vspace*{2mm} \\
\multicolumn{1}{l}{\hspace*{-.5cm}\mbox{\bf
gradlex:}} \\
  (e) >gl> (0)  & \Longrightarrow  & \deg(e)>0  \mbox { or } (e) >lex>
(0)\vspace*{2mm} \\
\multicolumn{1}{l}{\hspace*{-.5cm}\mbox{\bf
revgradlex:}}\\
  (e) >rgl> (0) & \Longrightarrow & \deg(e)>0  \mbox{ or }(e)  <lex<
(0)
\end{array}
\]

Note that the $lex$ ordering is identical to the standard \REDUCE \
kernel ordering, when $korder$ is set explicitly to the sequence of
variables.

\index{default ! term order}
$lex$ is the default term order mode in the $groebner$ package.

It is beyond the scope of this manual to discuss the functionality of
the term order modes. See \cite{Buchberger:88}.

The list of variables is declared as an optional parameter of the
$torder$ statement (see below). If this declaration is missing
or if the empty list has been used, the variables are extracted from
the expressions automatically and the \REDUCE \  system order defines
their sequence; this can be influenced by setting an explicit order
via the $korder$ statement.

The result of a Gr\"obner calculation is algebraically correct only
with respect to the term order mode and the variable sequence
which was in effect during the calculation. This is important if
several calls to the $groebner$ package are done with the result of the
first being the input of the second call. Therefore we recommend
that you declare the variable list and the order mode explicitly.
Once declared it remains valid until you enter a new $torder$
statement. The operator $gvars$ helps you extract the variables
from a given set of polynomials, if an automatic reordering has been selected.

\subsection{The Buchberger Algorithm}
\index{Buchberger's Algorithm}
The Buchberger algorithm of the package is based on {\sc
Gebauer/M\"oller} \cite{Gebauer:88}.
Extensions are documented in \cite{Melenk:88} and \cite{Giovini:91}.

\section{Loading of the Package}
The following command loads the package into
\REDUCE (this syntax may vary according to the implementation):
\begin{center}
load\_package groebner;
\end{center}

The package contains various operators, and switches for control
over the reduction process. These are discussed in the following.

\section{The Basic Operators}

\subsection{Term Ordering Mode}

\begin{description}
\ttindex{torder}
\item [{\it torder}]($vl$,$m$,$[p_1,p_2,\ldots]$);

where $vl$ is a variable list (or the empty list if
no variables are declared explicitly),
$m$ is the name of a term ordering mode $lex$, $gradlex$,
$revgradlex$ (or another implemented mode) and
$[p_1,p_2,\ldots]$ are additional parameters for the
term ordering mode (not needed for the basic modes).

$torder$ sets variable set and the term ordering mode.
The default mode is $lex$. The previous description is returned
as a list with corresponding elements. Such a list can
alternatively be passed as sole argument to $torder$.

If the variable list is empty or if the $torder$ declaration
is omitted, the automatic variable extraction is activated.

\ttindex{gvars}
\item[{\it gvars}] ({\it\{exp$1$, exp$2$, $ \ldots$, exp$n$\}});

 where $\{exp1, exp2, \ldots , expn\}$ is a list of expressions or
equations.

$gvars$ extracts from the expressions $\{exp1, exp2, \ldots , expn\}$
the kernels, which can play the role of variables for a Gr\"obner
calculation. This can be used e.g. in a $torder$ declaration.
\end{description}

\subsection{$groebner$: Calculation of a Gr\"obner Basis}
\begin{description}
\ttindex{groebner}
\item[{\it groebner}] $\{exp1, exp2, \ldots , expm\}; $

where $\{exp1, exp2, \ldots , expm\}$ is a list of
expressions or equations.

$groebner$ calculates the Gr\"obner basis of the given set of
expressions with respect to the current $torder$ setting.

The Gr\"obner basis $\{1\}$ means that the ideal generated by the
input polynomials is the whole polynomial ring, or equivalently, that
the input polynomials have no zeroes in common.

As a side effect, the sequence of variables is stored as a \REDUCE \  list
in the shared variable
\ttindex{gvarslast}
\begin{center}
$gvarslast$.
\end{center}

This is important if the variables are reordered because of optimization:
you must set them afterwards explicitly as the current variable sequence
if you want to use the Gr\"obner basis in the sequel, e.g. for a
$preduce$ call. A basis has the property ``Gr\"obner'' only with respect
to the variable sequences which had been active during its computation.
\end{description}

\example \index{Groebner package ! example}
\begin{verbatim}
   torder({},lex)$
   groebner{3*x**2*y + 2*x*y + y + 9*x**2 + 5*x - 3,
   2*x**3*y - x*y - y + 6*x**3 - 2*x**2 - 3*x + 3,
   x**3*y + x**2*y + 3*x**3 + 2*x**2 };

               2
     {8*x - 2*y  + 5*y + 3,

         3      2
      2*y  - 3*y  - 16*y + 21}
\end{verbatim}

This example used the default system variable ordering, which was
$\{x,y\}$. With the other variable ordering, a different basis results:

\begin{verbatim}
   torder({y,x},lex)$
   groebner{3*x**2*y + 2*x*y + y + 9*x**2 + 5*x - 3,
   2*x**3*y - x*y - y + 6*x**3 - 2*x**2 - 3*x + 3,
   x**3*y + x**2*y + 3*x**3 + 2*x**2 };

               2
     {2*y + 2*x  - 3*x - 6,

         3      2
      2*x  - 5*x  - 5*x}
\end{verbatim}

Another basis yet again results with a different term ordering:
\begin{verbatim}
   torder({x,y},revgradlex)$
   groebner{3*x**2*y + 2*x*y + y + 9*x**2 + 5*x - 3,
   2*x**3*y - x*y - y + 6*x**3 - 2*x**2 - 3*x + 3,
   x**3*y + x**2*y + 3*x**3 + 2*x**2 };

    2
{2*y  - 5*y - 8*x - 3,

 y*x - y + x + 3,

    2
 2*x  + 2*y - 3*x - 6}

\end{verbatim}

The operation of $groebner$ can be controlled by the following switches:
\begin{description}
\ttindex{groebopt}
\item[groebopt] -- If set $on$, the sequence of variables is optimized
with respect to execution speed; the algorithm involved is described
in~\cite{Boege:86}; note that the final list of variables is available in
\ttindex{gvarslast}
$gvarslast$.

An explicitly declared dependency supersedes the variable optimization. For example
\begin{center}
{\it depend} $a$, $x$, $y$;
\end{center}
guarantees that $a$ will be placed in front of $x$ and $y$. So
$groebopt$ can be used even in cases where elimination of variables is desired.

By default $groebopt$ is $off$, conserving the original variable sequence.

\ttindex{groebfullreduction}
\item[$groebfullreduction$] -- If set $off$, the reduction steps during
the \linebreak[4] $groebner$ operation are limited to the pure head
term reduction; subsequent terms are reduced otherwise.

By default $groebfullreduction$ is on.

\ttindex{gltbasis}
\item[$gltbasis$] -- If set on, the leading terms of the result basis are
extracted. They are collected in a basis of monomials, which is
available as value of the global variable with the name $gltb$.

\item[$glterms$] -- If $\{exp_1, \ldots , exp_m\} $ contain parameters
(symbols which are not member of the variable list), the share variable
{\tt $glterms$} contains a list of expression which during the
calculation were assumed to be nonzero. A Gr\"obner basis
is valid only under the assumption that all these expressions do not vanish.

\end{description}

The following switches control the print output of $groebner$; by
default all these switches are set $off$ and nothing is printed.
\begin{description}
\ttindex{groebstat}
\item[$groebstat$] -- A summary of the computation is printed
including the computing time, the number of intermediate
$h$--polynomials and the counters for the hits of the criteria.

\ttindex{trgroeb}
\item[$trgroeb$] -- Includes $groebstat$ and the printing of the
intermediate $h$-polynomials.

\ttindex{trgroebs}
\item[$trgroebs$] -- Includes $trgroeb$ and the printing of
intermediate $s$--poly\-nomials.

\ttindex{trgroeb1}
\item[$trgroeb1$] -- The internal pairlist is printed when modified.
\end{description}

\subsection{$Gzerodim$?: Test of $\dim = 0$}
\begin{description}
\ttindex{gzerodim?}
\item[{\it gzerodim}!?] $bas$ \\
where {\it bas} is a Gr\"obner basis in the current setting.
The result is {\it nil}, if {\it bas} is the
basis of an ideal of polynomials with more than finitely many common zeros.
If the ideal is zero dimensional, i. e. the polynomials of the ideal have only
finitely many zeros in common, the result is an integer $k$ which is the number
of these common zeros (counted with multiplicities).
\end{description}

\subsection{$gdimension$, $gindependent$\_$sets$: compute dimension and
independent variables}
The following operators can be used to compute the dimension
and the independent variable sets of an ideal which has the
Gr\"obner basis {\it bas} with arbitrary term order:
\begin{description}
\ttindex{gdimension}\ttindex{gindependent\_sets}
\ttindex{ideal dimension}\ttindex{independent sets}
\item[$gdimension$]$bas$
\item[$gindependent$\_$sets$]$bas$
{\it gindependent\_sets} computes the maximal
left independent variable sets of the ideal, that are
the variable sets which play the role of free parameters in the
current ideal basis. Each set is a list which is a subset of the
variable list. The result is a list of these sets. For an
ideal with dimension zero the list is empty.
{\it gdimension} computes the dimension of the ideal,
which is the maximum length of the independent sets.
\end{description}

The switch $groebopt$ plays no role in the algorithms $gdimension$ and
$gindependent$\_$sets$. It is set $off$ during the processing even if
it is set $on$ before. Its state is saved during the processing.

The ``Kredel-Weispfenning" algorithm is used (see \cite{Kredel:88a},
extended to general ordering in \cite{BeWei:93}.

\subsection{Conversion of a Gr\"obner Basis}

\subsubsection{$glexconvert$: Conversion of an Arbitrary Gr\"obner Basis
of a Zero Dimensional Ideal into a Lexical One}
\begin{description}
\ttindex{glexconvert}
\item[{\it glexconvert}] $ \left(\{exp,\ldots , expm\} \left[,\{var1
\ldots , varn\}\right]\left[,maxdeg=mx\right]\right.$ \\
$\left.\left[,newvars=\{nv1, \ldots , nvk\}\right]\right) $ \\
where $\{exp1, \ldots , expm\}$ is a Gr\"obner basis with
$\{var1, \ldots , varn\}$ as variables in the current term order mode,
$mx$ is an integer, and
$\{nv1, \ldots , nvk\}$ is a subset of the basis variables.
For this operator the source and target variable sets must be specified
explicitly.
\end{description}

$glexconvert$ converts a basis of a zero-dimensional ideal (finite number
of isolated solutions) from arbitrary ordering into a basis under {\it
lex} ordering. During the call of $glexconvert$ the original ordering of
the input basis must be still active!

$newvars$ defines the new variable sequence. If omitted, the
original variable sequence is used. If only a subset of variables is
specified here, the partial ideal basis is evaluated. For the
calculation of a univariate polynomial, $new$\-$vars$ should be a list
with one element.

$maxdeg$ is an upper limit for the degrees. The algorithm stops with
an error message, if this limit is reached.

A warning occurs if the ideal is not zero dimensional.

$glexconvert$ is an implementation of the FLGM algorithm by
\linebreak[4] {\sc Faug{\`e}re}, {\sc Gianni}, {\sc Lazard} and {\sc
Mora} \cite{Faugere:89}. Often, the calculation of a Gr\"obner basis
with a graded ordering and subsequent conversion to {\it lex} is
faster than a direct {\it lex} calculation. Additionally, $glexconvert$
can be used to transform a {\it lex} basis into one with different
variable sequence, and it supports the calculation of a univariate
polynomial. If the latter exists, the algorithm is even applicable in
the non zero-dimensional case, if such a polynomial exists.
If the polynomial does not exist, the algorithm computes  until $maxdeg$
has been reached.
\begin{verbatim}
   torder({{w,p,z,t,s,b},gradlex)

   g  :=  groebner  { f1 := 45*p + 35*s -165*b -36,
         35*p + 40*z + 25*t - 27*s, 15*w + 25*p*s +30*z -18*t
        -165*b**2, -9*w + 15*p*t  + 20*z*s,
        w*p + 2*z*t - 11*b**3, 99*w - 11*s*b +3*b**2,
        b**2 + 33/50*b + 2673/10000};

  g := {60000*w + 9500*b + 3969,

      1800*p - 3100*b - 1377,

      18000*z + 24500*b + 10287,

      750*t - 1850*b + 81,

      200*s - 500*b - 9,
             2
      10000*b  + 6600*b + 2673}

   glexconvert(g,{w,p,z,t,s,b},maxdeg=5,newvars={w});

               2
    100000000*w  + 2780000*w + 416421

   glexconvert(g,{w,p,z,t,s,b},maxdeg=5,newvars={p});

          2
    6000*p  - 2360*p + 3051

\end{verbatim}

\subsubsection{$groebner$\_$walk$: Conversion of a (General) Total Degree
Basis into a Lex One}
The algorithm $groebner$\_$walk$ convertes from an arbitrary polynomial
system a $graduated$ basis of the given variable sequence to a $lex$ one
of the same sequence. The job is done by computing a sequence
of Gr\"obner bases of correspondig monomial ideals, lifting the original
system each time. The algorithm has been described (more generally) by
\cite{AGK:961},\cite{AGK:962},\cite{AG:98} and \cite{CKM:97}.
$groebner\_walk$ should be only called, if the direct calculation of a
$lex$ Gr\"obner base does not work. The computation of $groebner\_walk$
includes some overhead (e. g. the computation divides polynomials).
Normally $torder$ must be called before to define the variables and the variable
sorting. The reordering of variables makes no sense with $groebner$\_$walk$;
so do not call $groebner\_walk$ with $groebopt$ $on$!

\begin{description}
\ttindex{groebner\_walk}
\item[{\it groebner\_walk}] $g$\\
where $g$ is a polynomial ideal basis computed under $gradlex$ or under
$weighted$ with a one--element, non zero weight vector with only one
element, repeated for each variable. The result is a corresponding
$lex$ basis (if that is computable), independet of the degree of the
ideal (even for non zero degree ideals).
The variabe $gvarslast$ is not set.
\end{description}

\subsection{$groebnerf$: Factorizing Gr\"obner Bases}

\subsubsection{Background}
If Gr\"obner bases are computed in order to solve systems of
equations or to find the common roots of systems of polynomials,
the factorizing version of the Buchberger algorithm can be used.
The theoretical background is simple: if a polynomial $p$ can be
represented as a product of two (or more) polynomials, e.g. $h= f*g$,
then $h$ vanishes if and only if one of the factors vanishes. So if
during the calculation of a Gr\"obner basis $h$ of the above form is
detected, the whole problem can be split into two (or more)
disjoint branches. Each of the branches is simpler than the complete
problem; this saves computing time and space. The result of this
type of computation is a list of (partial) Gr\"obner bases; the
solution set of the original problem is the union of the solutions of
the partial problems, ignoring the multiplicity of an individual
solution. If a branch results in a basis $\{1\}$, then there is no
common zero, i.e. no additional solution for the original problem,
contributed by this branch.

\subsubsection{$groebnerf$ Call}
\ttindex{groebnerf}
The syntax of $groebnerf$ is the same as for $groebner$.
\[ \mbox{\it groebnerf}(\{exp1, exp2, \ldots , expm\}
         [,\{\},\{nz1, \ldots nzk\}); \]
where $\{exp1, exp2, \ldots , expm\} $ is a given list of expressions or
equations, and $\{nz1, \ldots nzk\}$ is
an optional list of polynomials known to be non-zero.

$groebnerf$ tries to separate polynomials into individual factors and
to branch the computation in a recursive manner (factorization tree).
The result is a list of partial Gr\"obner bases. If no factorization can
be found or if all branches but one lead to the trivial basis $\{1\}$,
the result has only one basis; nevertheless it is a list of lists of
polynomials. If no solution is found, the result will be $\{\{1\}\}$.
Multiplicities (one factor with a higher power, the same partial basis
twice) are deleted as early as possible in order to speed up the
calculation. The factorizing is controlled by some switches.

As a side effect, the sequence of variables is stored as a \REDUCE \  list in
the shared variable
\begin{center}
gvarslast .
\end{center}
If $gltbasis$ is on, a corresponding list of leading term bases is
also produced and is available in the variable $gltb$.

The third parameter of $groebnerf$ allows one to declare some polynomials
nonzero. If any of these is found in a branch of the calculation
the branch is cancelled. This can be used to save a substantial amount
of computing time. The second parameter must be included as an
empty list if the third parameter is to be used.

\begin{verbatim}
   torder({x,y},lex)$
   groebnerf { 3*x**2*y + 2*x*y + y + 9*x**2 + 5*x = 3,
               2*x**3*y - x*y - y + 6*x**3 - 2*x**2 - 3*x = -3,
                x**3*y + x**2*y + 3*x**3 + 2*x**2 \};

       {{y - 3,x},

                      2
    {2*y + 2*x - 1,2*x  - 5*x - 5}}
\end{verbatim}

It is obvious here that the solutions of the equations can be read
off immediately.

All switches from $groebner$ are valid for $groebnerf$ as well:
\ttindex{groebopt}  \ttindex{gltbasis}
\ttindex{groebfullreduction} \ttindex{groebstat} \ttindex{trgroeb}
\ttindex{trgroebs} \ttindex{trgroeb1}
\begin{center}
\begin{tabular}{l}
$groebopt$ \\
$gltbasis$ \\
$groebfullreduction$ \\
$groebstat$ \\
$trgroeb$ \\
$trgroebs$ \\
$rgroeb1$
\end{tabular}
\end{center}

\subsubsection*{Additional switches for $groebnerf$:}
\begin{description}

\ttindex{trgroebr}
\item[$trgroebr$] -- All intermediate partial basis are printed when
detected.

By default $trgroebr$ is off.
\end{description}
{\it groebmonfac  groebresmax  groebrestriction} \\
\hspace*{.5cm} These variables are described in the following
paragraphs.

\subsubsection{Suppression of Monomial Factors}
The factorization in $groebnerf$ is controlled by the following
\ttindex{groebmonfac}
switches and variables.  The variable $groebmonfac$ is connected to
the handling of ``monomial factors''.  A monomial factor is a product
of variable powers occurring as a factor, e.g. $ x**2*y$  in  $x**3*y -
2*x**2*y**2$.  A monomial factor represents a solution of the type
``$ x = 0$  or  $y = 0$'' with a certain multiplicity.  With
$groeb$\-$nerf$ \ttindex{groebnerf}
the multiplicity of monomial factors is lowered to the value of the
shared variable
\ttindex{groebmonfac}
\begin{center}
$groebmonfac$
\end{center}
which by default is 1 (= monomial factors remain present, but their
multiplicity is brought down). With
\begin{center}
$groebmonfac$ := 0
\end{center}
the monomial factors are suppressed completely.

\subsubsection{Limitation on the Number of Results}
The shared variable
\ttindex{groebresmax}
\begin{center}
$groebresmax$
\end{center}
controls the number of partial results. Its default value is 300. If
$groebresmax$ partial results are calculated, the calculation is
terminated. $groebresmax$ counts all branches, including those which
are terminated (have been computed already), give no contribution to
the result (partial basis 1), or which are unified in the result with
other (partial) bases. So the resulting number may be much smaller.
When the limit of $groeresmax$ is reached, a warning

GROEBRESMAX limit reached

is issued; this warning in any case has to be taken as a serious one.
For "normal" calculations the $groebresmax$ limit is not reached.
$groebresmax$ is a shared variable (with an integer value); it can be
set in the algebraic mode to a different (positive integer) value.

\subsubsection{Restriction of the Solution Space}
In some applications only a subset of the complete solution set
of a given set of equations is relevant, e.g. only
nonnegative values or positive definite values for the variables.
A significant amount of computing time can be saved if
nonrelevant computation branches can be terminated early.

Positivity: If a polynomial has no (strictly) positive zero, then
every system containing it has no nonnegative or strictly positive
solution. Therefore, the Buchberger algorithm tests the coefficients of
the polynomials for equal sign if requested. For example, in $13*x +
15*y*z $ can be zero with real nonnegative values for $x, y$ and $z$
only if $x=0$ and $y=0$ or $ z=0$; this is a sort of ``factorization by
restriction''. A polynomial $13*x + 15*y*z + 20$ never can vanish
with nonnegative real variable values.

Zero point:  If any polynomial in an ideal has an absolute term, the ideal
cannot have the origin point as a common solution.

By setting the shared variable
\ttindex{groebrestriction}
\begin{center} $groebrestriction$ \end{center}
$groebnerf$ is informed of the type of restriction the user wants to
impose on the solutions:
\begin{center}
\begin{tabular}{l}
{\it groebrestiction:=nonnegative;} \\
\hspace*{+.5cm} only nonnegative real solutions are of
interest\vspace*{4mm} \\
{\it groebrestriction:=positive;} \\
\hspace*{+.5cm}only nonnegative and nonzero solutions are of
interest\vspace*{4mm} \\
{\it groebrestriction:=zeropoint;} \\
\hspace*{+.5cm}only solution sets which contain the point
$\{0,0,\ldots,0\}$ are or interest.
\end{tabular}
\end{center}

If $groebnerf$ detects a polynomial which formally conflicts with the
restriction, it either splits the calculation into separate branches, or,
if a violation of the restriction is determined, it cancels the actual
calculation branch.

\subsection{$greduce$, $preduce$: Reduction of Polynomials}

\subsubsection{Background} \label{groebner:background}
Reduction of a polynomial ``$p$'' modulo a given sets of polynomials
``$b$'' is done by the reduction algorithm incorporated in the
Buchberger algorithm. Informally it can be described for
polynomials over a field as follows:
\begin{center}
\begin{tabular}{l}
loop1: \hspace*{2mm}\% head term elimination \\
\hspace*{-1cm} if there is one polynomial $b$ in $B$ such that the
leading \\ term of $p$ is a multiple of the leading term of $P$ do \\
$p := p - lt(p)/lt(b) * b$  (the leading term vanishes)\\
\hspace*{-1cm} do this loop as long as possible; \\
loop2: \hspace*{2mm} \% elimination of subsequent terms \\
\hspace*{-1cm} for each term $s$ in $p$ do \\
if there is one polynomial $b$ in $B$ such that $s$ is a\\
multiple of the leading term of $p$ do \\
$p := p - s/lt(b) * b$ (the term $s$ vanishes) \\
\hspace*{-1cm}do this loop as long as possible;
\end{tabular}
\end{center}

If the coefficients are taken from a ring without zero divisors we
cannot divide by each possible number like in the field case. But
using that in the field case,  $c*p $ is reduced to  $c*q $, if $ p $
is reduced to $ q $, for arbitrary numbers $ c $,  the reduction for
the ring case uses the least $ c $ which makes the (field) reduction
for $ c*p $ integer. The result of this reduction is returned as
(ring) reduction of $ p $ eventually after removing the content, i.e.
the greatest common divisor of the coefficients. The result of this
type of reduction is also called a pseudo reduction of $ p $.

\subsubsection{Reduction via Gr\"obner Basis Calculation}
\ttindex{greduce}
\[
\mbox{\it greduce}(exp, \{exp1, exp2, \ldots , expm\}]);
\]
where {\it exp} is an expression, and $\{exp1, exp2,\ldots , expm\}$ is
a list of any number of expressions or equations.

$greduce$ first converts the list of expressions $\{exp1, \ldots ,
expn\}$ to a Gr\"obner basis, and then reduces the given expression
modulo that basis.  An error results if the list of expressions is
inconsistent. The returned value is an expression representing the
reduced polynomial. As a side effect, $greduce$ sets the variable {\it
gvarslast} in the same manner as $groebner$ does.

\subsubsection{Reduction with Respect to Arbitrary Polynomials}
\ttindex{preduce}
\[
 preduce(exp, \{exp1, exp2,\ldots , expm\});
\]
where $ expm $  is an expression, and $\{exp1, exp2, \ldots ,
expm \}$ is a list of any number of expressions or equations.

$preduce$ reduces the given expression modulo the set $\{exp1,
\ldots , expm\}$. If this set is a Gr\"obner basis, the obtained reduced
expression is uniquely determined. If not, then it depends on the
subsequence of the single reduction steps
(see~\ref{groebner:background}). $preduce$ does not check whether
$\{exp1, exp2, \ldots , expm\}$ is a Gr\"obner basis in the actual
order. Therefore, if the expressions are a Gr\"obner basis calculated
earlier with a variable sequence given explicitly or modified by
optimization, the proper variable sequence and term order must
be activated first.

\example ($preduce$ called with a Gr\"obner basis):
\begin{verbatim}
  torder({x,y},lex);
  gb:=groebner{3*x**2*y + 2*x*y + y + 9*x**2 + 5*x - 3,
               2*x**3*y - x*y - y + 6*x**3 - 2*x**2 - 3*x + 3,
               x**3*y + x**2*y + 3*x**3 + 2*x**2}$
  preduce (5*y**2 + 2*x**2*y + 5/2*x*y + 3/2*y
             + 8*x**2 + 3/2*x - 9/2, gb);

      2
     y
\end{verbatim}

\subsubsection{$greduce$\_$orders$: Reduction with several term orders}
The shortest polynomial with different polynomial term orders is computed
with the operator $greduce$\_$orders$:

\begin{description}
\ttindex{$greduce$\_$orders$}
\item[{\it greduce\_orders}]($exp$, \{$exp1$, $exp2$, \ldots , $expm$\}
[,\{$v_1$,$v_2$ \ldots $v_n$\}]);

where {\it exp} is an expression and $\{exp1, exp2,\ldots , expm\}$ is
a list of any number of expressions or equations. The list of variables
$v_1,v_2 \ldots v_n$ may be omitted; if set, the variables must be a list.
\end{description}

The expression {\it exp} is reduced by {\it greduce} with the orders
in the shared variable {\it gorders}, which must be a list of term
orders (if set). By default it is set to

\begin{center}
$\{revgradlex,gradlex,lex\}$
\end{center}

The shortest polynomial is the result.
The order with the shortest polynomial is set to the shared variable
{\it gorder}. A Gr\"obner basis of the system \{$exp1$, $exp2$, \ldots ,
$expm$\} is computed for each element of $orders$.
With the default setting {\it gorder} in most cases will be set
to {\it revgradlex}.
If the variable set is given, these variables are taken; otherwise all
variables of the system \{$exp1$, $exp2$, \ldots , $expm$\} are
extracted.

The Gr\"obner basis computations can take some time; if interrupted, the
intermediate result of the reduction is set to the shared variable
$greduce$\_$result$, if one is done already. However, this is not
nesessarily the minimal form.

If the variable {\it gorders} should be set to orders with a parameter,
the term oder has to be replaced by a list; the first element is the
term oder selected, followed by its parameter(s), e.g.

\begin{center}
$orders:=\{\{gradlexgradlex,2\},\{lexgradlex,2\}\}$
\end{center}

\subsubsection{Reduction Tree}
In some case not only are the results produced by $greduce$ and
$preduce$ of interest, but the reduction process is of some value
too. If the switch
\ttindex{groebprot}
\begin{center}
$groebprot$
\end{center}
is set on, $groebner$, $greduce$ and $preduce$ produce as a side effect
a trace of their work as a \REDUCE \  list of equations in the shared variable
\ttindex{groebprotfile}
\begin{center}
$groebprotfile$.
\end{center}
Its value is a list of equations with a variable ``candidate'' playing
the role of the object to be reduced. The polynomials are cited as
``$poly1$'', ``$poly2$'', $\ldots\;$. If read as assignments, these equations
form a program which leads from the reduction input to its result.
Note that, due to the pseudo reduction with a ring as the coefficient
domain, the input coefficients may be changed by global factors.

\newpage

\example \index{groebner package ! example}

{\it on groebprot} \$ \\
{\it preduce} $ (5*y**2 + 2*x**2*y + 5/2*x*y + 3/2*y + 8*x**2 $ \\
\hspace*{+1cm} $+ 3/2*x - 9/2, gb);$
\begin{verbatim}
      2
     y
\end{verbatim}
{\it groebprotfile;}
\begin{verbatim}
                  2         2                     2
    {candidate=4*x *y + 16*x  + 5*x*y + 3*x + 10*y  + 3*y - 9,

              2
     poly1=8*x - 2*y  + 5*y + 3,

              3      2
     poly2=2*y  - 3*y  - 16*y + 21,
     candidate=2*candidate,
     candidate= - x*y*poly1 + candidate,
     candidate= - 4*x*poly1 + candidate,
     candidate=4*candidate,

                   3
     candidate= - y *poly1 + candidate,
     candidate=2*candidate,

                     2
     candidate= - 3*y *poly1 + candidate,
     candidate=13*y*poly1 + candidate,
     candidate=candidate + 6*poly1,

                     2
     candidate= - 2*y *poly2 + candidate,
     candidate= - y*poly2 + candidate,
     candidate=candidate + 6*poly2}

 \end{verbatim}
This means
\begin{eqnarray*}
\lefteqn{
16 (5 y^2 + 2 x^2 y + \frac{5}{2} x y + \frac{3}{2} y
+ 8 x^2+ \frac{3}{2} x - \frac{9}{2})=} \\ & &
(-8 x y -32 x -2 y^3 -3 y^2 + 13 y + 6) \mbox{poly1} \\
& & \; + (-2 y^2 -2 y + 6) \mbox{poly2  } \; + y^2.
\end{eqnarray*}

\subsection{Tracing with $groebnert$ and $preducet$}
Given a set of polynomials $\{f_1,\ldots ,f_k\}$ and their Gr\"obner
basis $\{g_1,\ldots ,g_l\}$, it is well known that there are matrices of
polynomials $C_{ij}$ and $D_{ji}$ such that
\[
f_i = \displaystyle{\sum\limits_j} C_{ij} g_j \;\mbox{  and  } g_j =
\displaystyle{\sum\limits_i} D_{ji} f_i
\]
and these relations are needed explicitly sometimes.
In {\sc Buchberger} \cite{Buchberger:85}, such cases are described in the
context of linear polynomial equations. The standard technique for
computing the above formulae is to perform
Gr\"obner reductions, keeping track of the
computation in terms of the input data. In the current package such
calculations are performed with (an internally hidden) cofactor
technique: the user has to assign unique names to the input
expressions and the  arithmetic combinations are done with the
expressions and with their names simultaneously. So the result is
accompanied by an expression which relates it algebraically to the
input values.

\ttindex{groebnert} \ttindex{preducet}
There are two complementary operators with this feature: $groebnert$
and $preducet$; functionally they correspond to $groebner$ and $preduce$.
However, the sets of expressions here {\it {\bf must be}} equations
with unique single identifiers on their left side and the {\it lhs} are
interpreted as names of the expressions. Their results are
sets of equations ($groebnert$) or equations ($preducet$), where
a {\it lhs} is the computed value, while the {\it rhs} is its equivalent
in terms of the input names.

\example \index{groebner package ! example}

We calculate the Gr\"obner basis for an ellipse (named ``$p1$'' ) and a
line (named ``$p2$'' ); $p2$ is member of the basis immediately and so
the corresponding first result element is of a very simple form; the
second member is a combination of $p1$ and $p2$ as shown on the
{\it rhs} of this equation:

\begin{verbatim}
gb1:=groebnert {p1=2*x**2+4*y**2-100,p2=2*x-y+1};

gb1 := {2*x - y + 1=p2,
           2
        9*y  - 2*y - 199= - 2*x*p2 - y*p2 + 2*p1 + p2}
\end{verbatim}

\example \index{groebner package ! example}

We want to reduce the polynomial \verb+ x**2+ {\it  wrt}
the above Gr\"obner basis and need knowledge about the reduction
formula. We therefore extract the basis polynomials from $gb1$,
assign unique names to them (here $g1$, $g2$) and call $preducet$.
The polynomial to be reduced here is introduced with the name $Q$,
which then appears on the {\it rhs} of the result. If the name for the
polynomial is omitted, its formal value is used on the right side too.

\begin{verbatim}
  gb2 := for k := 1:length gb1 collect
        mkid(g,k) = lhs part(gb1,k)$
  preducet (q=x**2,gb2);

 - 16*y + 208= - 18*x*g1 - 9*y*g1 + 36*q + 9*g1 - g2
\end{verbatim}

This output means
\[
x^2 = (\frac{1}{2} x + \frac{1}{4} y - \frac{1}{4}) g1
 + \frac{1}{36} g2 + (-\frac{4}{9} y + \frac{52}{9}).
\]

\example \index{groebner package ! example}

If we reduce a polynomial which is member of the ideal, we
consequently get a result with {\it lhs} zero:
\begin{verbatim}
   preducet(q=2*x**2+4*y**2-100,gb2);

   0= - 2*x*g1 - y*g1 + 2*q + g1 - g2
\end{verbatim}

This means
\[ q = ( x + \frac{1}{2} y - \frac{1}{2}) g1 + \frac{1}{2} g2.
\]

With these operators the matrices $C_{ij}$ and $D_{ji}$ are available
implicitly, $D_{ji}$ as side effect of $groebnert$T, $c_{ij}$ by {\it calls}
of $preducet$ of $f_i$ {\it wrt} $\{g_j\}$. The latter by definition will
have the {\it lhs} zero and a {\it rhs} with linear $f_i$.

If $\{1\}$ is the Gr\"obner basis, the $groebnert$ calculation gives
a ``proof'', showing,  how  $1$ can be computed as combination of the
input polynomials.

\paragraph{Remark:} Compared to the non-tracing algorithms, these
operators are much more time consuming. So they are applicable
only on small sized problems.

\subsection{Gr\"obner Bases for Modules}

Given a polynomial ring, e.g. $r=z[x_1 \cdots x_k]$ and
an integer $n>1$: the vectors with $n$ elements of $r$
form a $module$ under vector addition (= componentwise addition)
and multiplication with elements of $r$. For a submodule
given by a finite basis a Gr\"obner basis
can be computed, and the facilities of the $groebner$ package
can be used except the operators $groebnerf$ and $groesolve$.

The vectors are encoded using auxiliary variables which represent
the unit vectors in the module. E.g. using ${v_1,v_2,v_3}$ the
module element $[x_1^2,0,x_1-x_2]$ is represented as
$x_1^2 v_1 + x_1 v_3 - x_2 v_3$. The use of ${v_1,v_2,v_3}$
as unit vectors is set up by assigning the set of auxiliary variables
to the share variable $gmodule$, e.g.
\begin{verbatim}
   gmodule := {v1,v2,v3};
\end{verbatim}
After this declaration all monomials built from these variables
are considered as an algebraically independent basis of a vector
space. However, you had best use them only linearly. Once $gmodule$
has been set, the auxiliary variables automatically will be
added to the end of each variable list (if they are not yet
member there).
Example:
\begin{verbatim}
   torder({x,y,v1,v2,v3},lex)$
   gmodule := {v1,v2,v3}$
   g:=groebner{x^2*v1 + y*v2,x*y*v1 - v3,2y*v1 + y*v3};

       2
g := {x *v1 + y*v2,

              2
      x*v3 + y *v2,

       3
      y *v2 - 2*v3,

      2*y*v1 + y*v3}

   preduce((x+y)^3*v1,g);

             1   3         2
 - x*y*v2 - ---*y *v3 - 3*y *v2 + 3*y*v3
             2

\end{verbatim}

In many cases a total degree oriented term order will be adequate
for computations in modules, e.g. for all cases where the
submodule membership is investigated. However, arranging
the auxiliary variables in an elimination oriented term order
can give interesting results. E.g.
\begin{verbatim}
   p1:=(x-1)*(x^2-x+3)$  p2:=(x-1)*(x^2+x-5)$
   gmodule := {v1,v2,v3};
   torder({v1,x,v2,v3},lex)$
   gb:=groebner {p1*v1+v2,p2*v1+v3};

gb := {30*v1*x - 30*v1 + x*v2 - x*v3 + 5*v2 - 3*v3,

        2       2
       x *v2 - x *v3 + x*v2 + x*v3 - 5*v2 - 3*v3}

   g:=coeffn(first gb,v1,1);

g := 30*(x - 1)

   c1:=coeffn(first gb,v2,1);

c1 := x + 5

   c2:=coeffn(first gb,v3,1);

c2 :=  - x - 3

   c1*p1 + c2*p2;

30*(x - 1)

\end{verbatim}
Here two polynomials
are entered as vectors $[p_1,1,0]$ and $[p_2,0,1]$. Using a term
ordering such that the first dimension ranges highest and the
other components lowest, a classical cofactor computation is
executed just as in the extended Euclidean algorithm.
Consequently the leading polynomial in the resulting
basis shows the greatest common divisor of $p_1$ and $p_2$,
found as a coefficient of $v_1$ while the coefficients
of $v_2$ and $v_3$ are the cofactors $c_1$ and $c_2$ of the polynomials
$p_1$ and $p_2$ with the relation $gcd(p_1,p_2) = c_1p_1 + c_2p_2$.

\subsection{Additional Orderings}
Besides the basic orderings, there are ordering options that are used for
special purposes.

\subsubsection{Separating the Variables into Groups }
\index{grouped ordering}
It is often desirable to separate variables
and formal parameters in a system of polynomials.
This can be done with a {\it lex} Gr\"obner
basis.  That however may be hard to compute as it does more
separation than necessary. The following orderings group the
variables into two (or more) sets, where inside each set a classical
ordering acts, while the sets are handled via their total degrees,
which are compared in elimination style. So the Gr\"obner basis will
eliminate the members of the first set, if algebraically possible.
{\it torder} here gets an additional parameter which describe the
grouping \ttindex{torder}
\begin{center}{\it
\begin{tabular}{l}
torder ($vl$,$gradlexgradlex$, $n$) \\
torder ($vl$,$gradlexrevgradlex$,$n$) \\
torder ($vl$,$lexgradlex$, $n$) \\
torder ($vl$,$lexrevgradlex$, $n$)
\end{tabular}}
\end{center}
Here the integer $n$ is the number of variables in the first group
and the names combine the local ordering for the first and second
group, e.g.
\begin{center}
\begin{tabular}{llll}
\multicolumn{4}{l}{{\it lexgradlex}, 3 for $\{x_1,x_2,x_3,x_4,x_5\}$:} \\
\multicolumn{4}{l}{$x_1^{i_1}\ldots x_5^{i_5} \gg x_1^{j_1}\ldots
x_5^{j_5}$} \\
if & & & $(i_1,i_2,i_3) \gg_{lex}(j_1,j_2,j_3)$ \\
& or & & $(i_1,i_2,i_3) = (j_1,j_2,j_3)$ \\
& & and & $(i_4,i_5) \gg_{gradlex}(j_4,j_5)$
\end{tabular}
\end{center}
Note that in the second place there is no {\it lex} ordering available;
that would not make sense.

\subsubsection{Weighted Ordering}
\ttindex{torder} \index{weighted ordering}
The statement
\begin{center}
\begin{tabular}{cl}
{\it torder} &($vl$,weighted, $\{n_1,n_2,n_3  \ldots$\}) ; \\
\end{tabular}
\end{center}
establishes a graduated ordering, where the exponents are first
multiplied by the given weights. If there are less weight values than
variables, the weight 1 is added automatically. If the weighted
degree calculation is not decidable, a $lex$ comparison follows.

\subsubsection{Graded Ordering}
\ttindex{torder} \index{graded ordering}
The statement
\begin{center}
\begin{tabular}{cl}
{\it torder} &($vl$,graded, $\{n_1,n_2,n_3 \ldots\}$,$order_2$) ; \\
\end{tabular}
\end{center}
establishes a graduated ordering, where the exponents are first
multiplied by the given weights. If there are less weight values than
variables, the weight 1 is added automatically. If the weighted
degree calculation is not decidable, the term order $order_2$ specified
in the following argument(s) is used.  The ordering $graded$ is designed
primarily for use with the operator $dd\_groebner$.

\subsubsection{Matrix Ordering}
\ttindex{torder} \index{matrix ordering}
The statement
\begin{center}
\begin{tabular}{cl}
{\it torder} &($vl$,matrix, $m$) ; \\
\end{tabular}
\end{center}
where $m$ is a matrix with integer elements and row length which
corresponds to the variable number. The exponents of each monomial
form a vector; two monomials are compared by multiplying their
exponent vectors first with $m$ and comparing the resulting vector
lexicographically. E.g. the unit matrix establishes the classical
$lex$ term order mode, a matrix with a first row of ones followed
by the rows of a unit matrix corresponds to the $gradlex$ ordering.

The matrix $m$ must have at least as many rows as columns; a non--square
matrix contains redundant rows. The matrix must have full rank, and
the top non--zero element of each column must be positive.

The generality of the matrix based term order has its price: the
computing time spent in the term sorting is significantly higher
than with the specialized term orders. To overcome this problem,
you can compile a matrix term order ; the
compilation reduces the computing time overhead significantly.
If you set the switch $comp$ on, any new order matrix is compiled
when any operator of the $groebner$ package accesses it for the
first time. Alternatively you can compile a matrix explicitly
\begin{verbatim}
    torder_compile(<n>,<m>);
\end{verbatim}
where $<n>$ is a name (an identifier) and $<m>$ is a term order matrix.
$torder\_compile$ transforms the matrix into a LISP program, which
is compiled by the LISP compiler when $comp$ is on or when you
generate a fast loadable module. Later you can activate the new term
order by using the name $<n>$ in a $torder$ statement as term ordering
mode.

\subsection{Gr\"obner Bases for Graded Homogeneous Systems}

For a homogeneous system of polynomials under a term order
{\it graded}, {\it gradlex}, {\it revgradlex} or {\it weighted}
a Gr\"obner Base can be computed with limiting the grade
of the intermediate $s$--polynomials:
\begin{description}
\ttindex{dd\_groebner}
\item [{\it dd\_groebner}]($d1$,$d2$,$\{p_1,p_2,\ldots\}$);
\end{description}
where $d1$ is a non--negative integer and $d2$ is an integer
$>$ $d1$ or ``infinity". A pair of polynomials is considered
only if the grade of the lcm of their head terms is between
$d1$ and $d2$. See \cite{BeWei:93} for the mathematical background.
For the term orders {\it graded} or {\it weighted} the (first) weight
vector is used for the grade computation. Otherwise the total
degree of a term is used.

\section{Ideal Decomposition \& Equation System Solving}
Based on the elementary Gr\"obner operations, the $groebner$ package offers
additional operators, which allow the decomposition of an ideal or of a
system of equations down to the individual solutions.

\subsection{Solutions Based on Lex Type Gr\"obner Bases}

\subsubsection{groesolve: Solution of a Set of Polynomial Equations}
\ttindex{groesolve} \ttindex{groebnerf}
The $groesolve$ operator incorporates a macro algorithm;
lexical Gr\"obner bases are computed by $groebnerf$ and decomposed
into simpler ones by ideal decomposition techniques; if algebraically
possible, the problem is reduced to univariate polynomials which are
solved by $solve$; if $rounded$ is on, numerical approximations are
computed for the roots of the univariate polynomials.
\[
 groesolve(\{exp1, exp2, \ldots , expm\}[,\{var1, var2, \ldots ,
varn\}]); \]
where $\{exp1, exp2,\ldots , expm\}$ is a list of any number of
expressions or equations, $\{var1, var2, \ldots , varn\}$ is an
optional list of variables.

The result is a set of subsets. The subsets contain the solutions of the
polynomial equations. If there are only finitely many solutions,
then each subset is a set of expressions of triangular type
$\{exp1, exp2,\ldots , expn\},$ where $exp1$ depends only on
$var1,$ $exp2$ depends only on $var1$ and $var2$ etc. until $expn$ which
depends on $var1,\ldots,varn.$ This allows a successive determination of
the solution components. If there are infinitely many solutions,
some subsets consist in less than $n$ expressions. By considering some
of the variables as ``free parameters'',  these subsets are usually
again of triangular type.

\example (Intersections of a line with a circle):
\index{groebner package ! example}

\[ groesolve(\{x**2 - y**2 - a, p*x+q*y+s\},\{x,y\}); \]

\begin{verbatim}
                   2      2    2             2    2
   {{x=(sqrt( - a*p  + a*q  + s )*q - p*s)/(p  - q ),
                      2      2    2             2    2
     y= - (sqrt( - a*p  + a*q  + s )*p - q*s)/(p  - q )},
                      2      2    2             2    2
    {x= - (sqrt( - a*p  + a*q  + s )*q + p*s)/(p  - q ),
                   2      2    2             2    2
     y=(sqrt( - a*p  + a*q  + s )*p + q*s)/(p  - q )}}
\end{verbatim}

If the system is zero--dimensional (has a number of isolated solutions),
the algorithm described in \cite{Hillebrand:99} is used, if the decomposition
leaves a polynomial with mixed leading term. Hillebrand has written the
article and M\"oller was the tutor of this job.

The reordering of the $groesolve$ variables is controlled by the
\REDUCE \ switch $varopt$. If $varopt$ is $on$ (which is the default
of $varopt$), the variable sequence is optimized (the variables are reordered).
If $varopt$ is $off$, the given variable sequence is taken (if no variables
are given, the order of the \REDUCE \ system is taken instead). In general, the
reordering of the variables makes the Gr\"obner basis computation
significantly faster.
A variable dependency, declare by one (or several) $depend$ statements,
is regarded (if $varopt$ is $on$). The switch $groebopt$ has no meaning
for $groesolve$; it is stored during its processing.

\subsubsection{$groepostproc$: Postprocessing of a Gr\"obner Basis}
\ttindex{groepostproc}
In many cases, it is difficult to do the general Gr\"obner processing.
If a Gr\"obner basis with a {\it lex} ordering is calculated already (e.g.,
by very individual parameter settings), the solutions can be derived
from it by a call to $groepostproc$. $groesolve$ is functionally
equivalent to a call to $groebnerf$ and subsequent calls to
$groepostproc$ for each partial basis.
\[
 groepostproc(\{exp1, exp2, \ldots , expm\}[,\{var1, var2, \ldots ,
varn\}]);
\]
where $\{exp1, exp2, \ldots , expm\}$ is a list of any number of
expressions, \linebreak[4] $\{var1, var2, \ldots ,$ $ varn\}$ is an
optional list of variables. The expressions must be a {\it lex} Gr\"obner
basis with the given variables; the ordering must be still active.

The result is the same as with $groesolve$.

\begin{verbatim}
groepostproc({x3**2 + x3 + x2 - 1,
              x2*x3 + x1*x3 + x3 + x1*x2 + x1 + 2,
              x2**2 + 2*x2 - 1,
              x1**2 - 2},{x3,x2,x1});

{{x3= - sqrt(2),

  x2=sqrt(2) - 1,

  x1=sqrt(2)},

 {x3=sqrt(2),

  x2= - (sqrt(2) + 1),

  x1= - sqrt(2)},

      sqrt(4*sqrt(2) + 9) - 1
 {x3=-------------------------,
                 2

  x2= - (sqrt(2) + 1),

  x1=sqrt(2)},

       - (sqrt(4*sqrt(2) + 9) + 1)
 {x3=------------------------------,
                   2

  x2= - (sqrt(2) + 1),

  x1=sqrt(2)},

      sqrt( - 4*sqrt(2) + 9) - 1
 {x3=----------------------------,
                  2

  x2=sqrt(2) - 1,

  x1= - sqrt(2)},

       - (sqrt( - 4*sqrt(2) + 9) + 1)
 {x3=---------------------------------,
                     2

  x2=sqrt(2) - 1,

  x1= - sqrt(2)}}
\end{verbatim}

\subsubsection{Idealquotient: Quotient of an Ideal and an Expression}
\ttindex{idealquotient} \index{ideal quotient}
Let $i$ be an ideal and $f$ be a polynomial in the same
variables. Then the algebraic quotient is defined by
\[
i:f = \{ p \;| \; p * f \;\mbox{    member of }\; i\}\;.
\]
The ideal quotient $i:f$ contains $i$ and is obviously part of the
whole polynomial ring, i.e. contained in $\{1\}$. The case $i:f =
\{1\}$ is equivalent to $f$ being a member of  $i$. The other extremal
case, $i:f=i$, occurs, when $f$ does not vanish at any general zero of $i$.
The explanation of the notion ``general zero'' introduced by van der
Waerden, however, is beyond the aim of this manual. The operation
of $groesolve$/$groepostproc$ is based on nested ideal quotient
calculations.

If $i$ is given by a basis and $f$ is given as an expression, the
quotient can be calculated by
\[
idealquotient (\{exp1, \ldots , expm\}, exp); \]
where $\{exp1, exp2, \ldots , expm\}$ is a list of any number of
expressions or equations, {\it exp} is a single expression or equation.

$idealquotient$ calculates the algebraic quotient of the ideal $i$
with the basis  $\{exp1, exp2, \ldots , expm\}$ and {\it exp} with
respect to  the variables given or extracted.  $\{exp1, exp2, \ldots ,
expm\}$ is not necessarily a Gr\"obner basis.
The result is the Gr\"obner basis of the quotient.

\subsubsection{Saturation: Saturation of an Ideal and an Expression}
\ttindex{saturation}
The $saturation$ computes the quotient on an ideal and an arbitrary power
of an expression $exp**n$ with arbitrary $n$. The call is
\[ saturation (\{exp1, \ldots , expm\}, exp); \]
where $\{exp1, exp2, \ldots , expm\}$ is a list of any number of
expressions or equations, {\it exp} is a single expression or equation.

$saturation$ calls $idealquotient$ several times, until the result is
stable, and returns it.

\subsection{Operators for Gr\"obner Bases in all Term Orderings}
\index{Hilbert polynomial}
In some cases where no Gr\"obner
basis with lexical ordering can be calculated, a calculation with a total
degree ordering is still possible. Then the Hilbert polynomial gives
information about the dimension of the solutions space and for finite
sets of solutions univariate polynomials can be calculated. The solutions
of the equation system then is contained in the cross product of all
solutions of all univariate polynomials.

\subsubsection{Hilbertpolynomial: Hilbert Polynomial of an Ideal}
\ttindex{Hilbertpolynomial}
This algorithm was contributed by {\sc Joachim Hollman}, Royal
Institute of Technology, Stockholm (private communication).

\[
hilbertpolynomial (\{exp1, \ldots , expm\})\;;
\]
where $\{exp1, \ldots , expm\}$ is a list of any number of expressions
or equations.

$hilertpolynomial$ calculates the Hilbert polynomial of the ideal
with basis $\{exp1, \ldots , expm\}$ with respect to the
variables given or extracted provided the given term ordering is
compatible with the degree, such as the $gradlex$- or $revgradlex$-ordering.
The term ordering of the basis
must be active and $\{exp1, \ldots$, $ expm\}$ should be a
Gr\"obner basis with respect to this ordering. The Hilbert polynomial
gives information about the cardinality of solutions of the system
$\{exp1, \ldots , expm\}$: if the Hilbert polynomial is an
integer, the system has only a discrete set of solutions and the
polynomial is identical with the number of solutions counted with
their multiplicities. Otherwise the degree of the Hilbert
polynomial is the dimension of the solution space.

If the Hilbert polynomial is not a constant, it is constructed with the
variable ``x'' regardless of whether $x$ is member of
$\{var1, \ldots , varn\}$ or not. The value of this polynomial at
sufficiently large numbers  ``x'' is the difference
of the dimension of the linear vector space of all polynomials of degree
$ \leq x $ minus the dimension of the subspace of all polynomials of
degree $\leq x $ which belong also to the ideal.

$x$ must be an undefined variable or the value of $x$ must be an undefined
variable; otherwise a warning is given and a new (generated) variable
is taken instead.

\paragraph{Remark:} The number of zeros in an ideal and the
Hilbert polynomial depend only on the leading terms of the
Gr\"obner basis. So if a subsequent Hilbert calculation is planned, the
Gr\"obner calculation should be performed with $on$ $gltbasis$ and
the value of $gltb$ (or its elements in a $groebnerf$ context) should be
given to $hilbertpolynomial$. In this manner, a lot of computing time can be
saved in the case of long calculations.

\section{Calculations ``by Hand''}
The following operators support explicit calculations with
polynomials in a distributive representation at the \REDUCE \  top level.
So they allow one to do Gr\"obner type evaluations stepwise by
separate calls. Note that the normal \REDUCE \  arithmetic can be used
for arithmetic combinations of monomials and polynomials.

\subsection{Representing Polynomials in Distributive Form}
\ttindex{gsort}
\[ gsort p; \]
where $p$ is a polynomial or a list of polynomials.

If $p$ is a single polynomial, the result is a reordered version of $p$
in the distributive representation according to the variables and the
current term order mode; if $p$ is a list, its members are converted
into distributive representation and the result is the list sorted by
the term ordering of the leading terms; zero polynomials are
eliminated from the result.

\begin{verbatim}
     torder({alpha,beta,gamma},lex);

     dip := gsort(gamma*(alpha-1)**2*(beta+1)**2);

                2     2                2
    dip := alpha *beta *gamma + 2*alpha *beta*gamma

           2                     2
    + alpha *gamma - 2*alpha*beta *gamma - 4*alpha*beta*gamma

                           2
     - 2*alpha*gamma + beta *gamma + 2*beta*gamma + gamma

 \end{verbatim}

\subsection{Splitting of a Polynomial into Leading Term and Reductum}
\ttindex{gsplit}
\[ gsplit p; \]
where $p$ is a polynomial.

$gsplit$ converts the polynomial $p$ into distributive representation
and splits it into leading monomial and reductum. The result is a list
with two elements, the leading monomial and the reductum.

\begin{verbatim}
   gslit dip;

          2     2
    {alpha *beta *gamma,

            2                   2                     2
     2*alpha *beta*gamma + alpha *gamma - 2*alpha*beta *gamma

                         2
     - 4*alpha*beta*gamma - 2*alpha*gamma + beta *gamma

     + 2*beta*gamma + gamma}

 \end{verbatim}

\subsection{Calculation of Buchberger's S-polynomial}
\ttindex{gspoly}
\[ gspoly (p1,p2); \]
where $p1$  and $p2$ are polynomials.

$gspoly$ calculates the $s$-polynomial from $p1$  and $p2$;

Example for a complete calculation (taken from {\sc Davenport et al.}
 \cite{Davenport:88a}):
\begin{verbatim}
   torder({x,y,z},lex)$
   g1  :=  x**3*y*z - x*z**2;
   g2  :=  x*y**2*z - x*y*z;
   g3  :=  x**2*y**2 - z;$

   % first S-polynomial

   g4  :=  gspoly(g2,g3);$

           2        2
    g4 := x *y*z - z

    % next S-polynomial

    p :=  gspoly(g2,g4); $

          2          2
    p := x *y*z - y*z

    % and reducing, here only by g4

    g5  :=  preduce(p,{g4});

                2    2
    g5 :=  - y*z  + z

    % last S-polynomial}

    g6  :=  gspoly(g4,g5);

           2  2    3
    g6 := x *z  - z

    % and the final basis sorted descending

    gsort{g2,g3,g4,g5,g6};

      2  2
    {x *y  - z,

      2        2
     x *y*z - z ,

      2  2    3
     x *z  - z ,

        2
     x*y *z - x*y*z,

           2    2
      - y*z  + z }
 \end{verbatim}

\bibliography{groebner}
\bibliographystyle{plain}
\end{document}


\section{Methodology}
\label{methodology}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Dataset}
In this work, our previously generated dataset~\cite{Ijjeh2021} was used for training various deep learning models.
Further, the generated dataset resembles measurements acquired by SLDV in the transverse direction (perpendicular to the plate surface).
The dataset contains 475 simulated cases of full wavefield of propagating Lamb waves in a plate made of carbon fibre-reinforced polymer (CFRP).
Further, the simulated cases represent the interaction of Lamb waves with different delamination scenarios.
In which, each simulated case in the dataset computed from the wave propagation model is a 3D matrix, which holds the amplitudes of the propagating waves at location \((x,y)\) and time \((t)\).
Hence, these matrices can be seen as animated frames at discrete time \(t_k\) of propagating waves.
Moreover, it should be mentioned that the simulated delaminations were located closer to the top surface of the plate.
Accordingly, it is easier to detect delamination by using the simulated full wavefield on the top surface of the plate instead of the bottom surface.
However, to train our various models we applied the difficult case in which the full wavefield was registered at the bottom surface of the plate.
To enhance the visualisation, the root mean square depicted in Eq.~\ref{ref:rms} was applied for the full wavefield frames.
\begin{equation}
	\hat{s}(x,y) = \sqrt{\frac{1}{N}\sum_{k=1}^{N} s(x,y,t_k)^2}
	\label{ref:rms}
\end{equation}
where \(N\) refers to the number of sampling points 512,  and \((x,y)\) refers to the location.
Figures~\ref{fig:rmstop} and \ref{fig:rmsbottom} show the result of applying RMS to the full wavefield from the top and bottom surface of the plate respectively.
The dataset consisting of RMS images which were used in this research paper is available online~\cite{Kudela2020d}.
\begin{figure} [h!]
	\centering
	\begin{subfigure}[b]{0.47\textwidth}
		\centering
		\includegraphics[scale=.29]{RMS_flat_shell_Vz_27_500x500top.png}
		\caption{top}
		\label{fig:rmstop}
	\end{subfigure}
	\hfill
	\begin{subfigure}[b]{0.47\textwidth}
		\centering
		\includegraphics[scale=.29]{RMS_flat_shell_Vz_27_500x500bottom.png}
		\caption{bottom}
		\label{fig:rmsbottom}
	\end{subfigure}
	\caption{RMS of the full wavefield from the top surface of the plate (a) and the bottom surface of the plate (b).}
\label{fig:rms}
\end{figure}
\subsection{Data preprocessing}
To enhance the performance of the optimizer during the training process, the colour scale values were normalized to a range of (\(0-1\)) instead of the initial scale which was in a range of (\(0 - 255\)).
Furthermore, we have applied data augmentation on the dataset by flipping the images horizontally, vertically and diagonally.
As a result, the dataset size increased four times -- \(1900\)  images were produced.
We have split the dataset into two portions:  \(80\%\) for the training set and \(20\%\) for the testing set.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Moreover, a cross-validation (CV) method was applied to the training set to reduce the overfitting which happens when the model is able to fit on the training data, while it poorly fit on the new unseen data.
In other words, the model only learns the patterns of the training data therefore the model will not generalise well.
%Figure.~\ref{fig:Cross_validation} illustrates the K-fold CV technique.
%In this technique, we have split the training set into \(K\) small sets (folds), hence the name K-folds.
%Therefore, we iterate over the training set K iterations.
%During each iteration, the model uses  \(K-1\) folds for training and the remaining fold is used for validation.
%In our models, we have chosen \(K=5\), therefore, we have \(5\) iterations of training.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%For each iteration, we compute the performance of the model.
%Finally, we compute the cross-validation performance for all iterations as illustrated in Eqn.~(\ref{eq:cv_performance}) as a mean value over the K performance estimations of the validation fold set.
The main advantage of the K-fold CV method versus a regular train/test split is to reduce the overfitting by utilising data more efficiently as every data sample is used in both training and validation.
Therefore, by using this technique, we aim to improve the ability of the model to generalise and reduce the overfitting.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{equation}
%Final \ Performance = \frac{1}{K}\sum_{i=1}^{K}Performace
%\label{eq:cv_performance}
%\end{equation}
%\begin{figure}
%	\centering
%	\includegraphics[scale=1.0]{cross_validation.png}
%	\caption{K-fold Cross validation, K=\(5\).}
%	\label{fig:Cross_validation}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Introduction}

%\begin{itemize}
%\item Tuning is important in deep learning and tuning is hard.
%\item People do (a) adaptive (b) method with knob. (Fix momentum and only investigate lr).
%\item momentum accelerates. better understanding the behavior of momentum is interesting better tuner with less tuning. We revisit mom sgd and found the robustness to lr and variation. As well as related to async. And derive simple theory to analyze.
%\item These properties makes mom sgd a good candidate for auto tuning. We investigate a quadratic model and propose an simple and easy to understand tuner. It is faster than Adam
%\item We also connect to async theory and develope component and better than others in asynchrony.
%\end{itemize}

\outline{[Problem.]}
%Deep learning involves the use of very large datasets to fit large, complex models.
Accelerated forms of stochastic gradient descent (SGD), pioneered by
\citet{polyak1964some} and \citet{nesterov1983method}, are the de-facto
training algorithms for deep learning.
Their use requires a sane choice for their {\em hyperparameters}:
typically a {\em learning rate} and {\em momentum parameter} \citep{sutskever2013importance}.
\outline{[Hardness.]}
%Hyperparameter tuning is often cited as being the number-one factor in extracting good performance from deep learning systems.
%It is by far the most time consuming phase of developing a new deep learning model or system.
%One could say that tuning is {\em the hidden cost of modern machine learning},
However, tuning hyperparameters is arguably the most time-consuming part of deep learning, with thousands of grad-student-hours sacrificed and many papers outlining best tuning practices written
\cite{bengio2012practical,orr2003neural,bengio2012deep,bottou2012stochastic}.

\outline{[previous approach]}
Deep learning researchers have proposed a number of methods to deal with hyperparameter optimization.
Na\"ive methods, like the grid-search,
are prohibitively expensive for all but the smallest problems.
Smart black-box methods \cite{bergstra2012random,snoek2012practical}
do not explicitly take into account the problem specifics and spend time testing multiple configurations.
Adaptive methods provide an attractive alternative.
They aim to tune a single run on the fly and have been largely successful in relieving practitioners of tuning the learning rate.
Algorithms like Adagrad \cite{duchi2011adaptive}, RMSProp \cite{tieleman2012lecture} and Adam \cite{kingma2014adam} use the magnitude of gradient elements to tune learning rates {\em individually for each variable}. A common limitation of state-of-the-art adaptive methods is that they do not tune their momentum.
%Finally, methods like the one proposed by \citet{schaul2013no} use simple models and simple measurements to tune a global learning rate for the standard SGD update.

\outline{[limitation of previous approach]}
Momentum is a fundamental parameter at the heart of the acceleration process, lending its name to the most ubiquitous accelerated method \cite{polyak1964some}, simply dubbed {\em momentum}.
Classic \cite{polyak1964some} and recent results \cite{sutskever2013importance} alike show that proper momentum tuning has a significant impact on training speed.
%Unfortunately, there exists no method that automatically tunes its momentum parameter.
%Large-scale systems pose extra tuning challenges.
\jianedits{Momentum becomes even more critical on distributed systems.
Recently, \citet{mitliagkas2016asynchrony} showed that, in asynchronous-parallelization~\cite{recht2011hogwild,dean2012large,chilimbi2014project,hadjis2016omnivore},
a technique for efficient distributed training without synchronization locks,
%introduces additional momentum-like dynamics to momentum SGD, and
one can manually reduce algorithmic momentum to accelerate convergence~\cite{hadjis2016omnivore}.
As part of a collaboration with an industry affiliate and a big research lab, we verify that tuning momentum dramatically improves convergence on thousand-node scales.
However, grid-searching momentum on very large cluster jobs becomes especially challenging.
We believe that better understanding of momentum and its rich properties is interesting in its own right and could yield a next generation of adaptive methods that perform {\em automatic momentum tuning}. }
%Increasing model and dataset sizes motivate parallelization \cite{dean2012large,chilimbi2014project,hadjis2016omnivore,chen2016revisiting}.
% Asynchronous methods \cite{recht2011hogwild} provide efficient parallelization by training without locking or synchronization.
%The exact effect of asynchrony on the optimization process has been a mystery, though empirical results have shown great successes \cite{recht2011hogwild,dean2012large,chilimbi2014project,hadjis2016omnivore}.
%Recent work by \citet{mitliagkas2016asynchrony} reveals that asynchrony introduces additional momentum-like dynamics to the optimization process,
%a statement that can be made {\em exact} for the momentum SGD algorithm.
%\jianedits{Recent work by \citet{mitliagkas2016asynchrony} reveals that asynchrony introduces additional momentum-like dynamics to momentum SGD, and one can manually reduce algorithmic momentum to accelerate convergence~\cite{hadjis2016omnivore}}.

%As part of a collaboration with an industry affiliate and a big research lab, we had the opportunity to verify this phenomenon on thousand-node scales.
%Tuning the momentum value improves convergence speed,
%however grid-searching momentum values at that scale becomes challenging.
%Recent work \cite{hadjis2016omnivore} uses this theoretical understanding to perform targeted, efficient hyperparameter searches.
%We believe that better understanding of momentum and its rich properties is interesting in its own right and could yield a next generation of adaptive methods with {\em automatic momentum tuning}.

%%%%%%%% old un smoothed version %%%%%%%%%%%%%%%%%%
%%Large-scale systems pose extra tuning challenges.
%Momentum tuning becomes even more critical on large-scale, parallel systems.
%%Increasing model and dataset sizes motivate parallelization \cite{dean2012large,chilimbi2014project,hadjis2016omnivore,chen2016revisiting}.
% Asynchronous methods \cite{recht2011hogwild} provide efficient parallelization by training without locking or synchronization.
%The exact effect of asynchrony on the optimization process has been a mystery, though empirical results have shown great successes \cite{recht2011hogwild,dean2012large,chilimbi2014project,hadjis2016omnivore}.
%%Recent work by \citet{mitliagkas2016asynchrony} reveals that asynchrony introduces additional momentum-like dynamics to the optimization process,
%%a statement that can be made {\em exact} for the momentum SGD algorithm.
%\jianedits{Recent work by \citet{mitliagkas2016asynchrony} reveals that asynchrony introduces additional momentum-like dynamics to momentum SGD, and one can manually reduce algorithmic momentum to accelerate convergence~\cite{hadjis2016omnivore}}.
%
%As part of a collaboration with an industry affiliate and a big research lab, we had the opportunity to verify this phenomenon on thousand-node scales.
%Tuning the momentum value improves convergence speed,
%however grid-searching momentum values at that scale becomes challenging.
%Recent work \cite{hadjis2016omnivore} uses this theoretical understanding to perform targeted, efficient hyperparameter searches.
%We believe that better understanding of momentum and its rich properties is interesting in its own right and could yield a next generation of adaptive methods that also perform {\em automatic momentum tuning}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{wrapfigure}[11]{R}{0.46\textwidth}
\vspace{-1.0em}
\begin{minipage}{1.0\linewidth}
\begin{figure}[H]
	\includegraphics[width=1.0\linewidth]{experiment_results/spotlight.pdf}
	\caption{\tuner comparing to Adam.}
\end{figure}
\end{minipage}
\end{wrapfigure}
We revisit the basic SGD update that uses Polyak's momentum and a single learning rate for all variables.
We empirically show that, when hand-tuned, momentum SGD achieves faster convergence than Adam for a large class of models.
We then formulate the optimization update as a dynamical system and study certain robustness properties of the momentum operator.
Building on our analysis, we design \tuner, an automatic hyperparameter tuner for momentum SGD.
\tuner tunes the learning rate and momentum on the fly, and uses a novel {\em closed-loop control architecture} to compensate for the added dynamics of asynchrony.
Specifically:
\begin{itemize}[leftmargin=2em]
\item
The momentum operator's spectral radius is constant in a large subset of the hyperparameter space we call {\em the robust region}.
This is a known, but relatively obscure property.
Our analysis in Section~\ref{sec:momentum_operator} gives a novel interpretation:
momentum is robust to learning rate misspecification and curvature variation,
two desirable properties for deep learning.
%\item
%We analyze the momentum operator in Section~\ref{sec:momentum_operator} with focus on its spectral radius. More specifically, the spectral radius can be constant under conditions.
%We expose previously unknown insights from the simple property:
%the momentum operator can yield constant convergence rates even when the learning rate is grossly mis-tuned.
%Furthermore, constant convergence rates are empirically possible for some non-convex objectives,
%where curvature dramatically varies spatially.%  (a factor of more than $1000\times$ when momentum is set to $0.9$).
\item
In Section~\ref{sec:sync_tuner}, we use these insights and a simple quadratic model analysis to design \tuner, an automatic tuner for momentum SGD.
\tuner uses on-the-fly measurements from the gradients of the system to tune both learning rate and momentum.
%\item We use these insights and the analysis of a quadratic model to design
%\tuner in Section~\ref{sec:sync_tuner}. To tune the learning rate and momentum, the tuner minimize the squared distance from the minimum of a local quadratic approximation at each step of the algorithm.
%\item In Section~\ref{sec:async_tuner}, we present the first asynchrony-aware tuner.
%We first provide a measurement component to estimate the total amount of momentum in a running system, including the extra momentum due to asynchrony.
%Then, we propose \asynctuner, a closed-loop version of \tuner, that uses the measurement component and a negative feedback loop to bring the total amount of momentum to the levels suggested by the tuning rule.
%\item
%Furthermore, \tuner is {\em asynchrony-aware}.
%It uses a {\em novel momentum-measurement mechanism} that is part of a {\em closed-loop} design to automatically tune the value of momentum when asynchrony introduces extra momentum dynamics.
\item In Section~\ref{sec:async_tuner}, we present \asynctuner
%an asynchrony-aware tuner.
\jianedits{for asynchronous training}.
\jianedits{It} measures the total momentum in a running system, including any asynchrony-induced momentum.
This measurement is used in a negative feedback loop to control the value of algorithmic momentum.% on the fly.
%With the estimates, it use a negative feedback loop to bring the total momentum to the level suggested by \asynctuner.

%\item We present an analysis of the momentum operator in Section~\ref{sec:momentum_operator}.
%We focus on the fact that its spectral radius is constant in a certain robust region.
%We expose previously unknown insights that stem from that simple property:
%the momentum operator can yield constant convergence rates even when the learning rate is grossly mis-tuned.
%Furthermore, constant convergence rates are possible for certain non-convex objectives,
%where curvature can vary dramatically (a factor of more than $1000\times$ when momentum is set to $0.9$).
\end{itemize}

%Asynchronous methods \cite{recht2011hogwild} provide very efficient parallelization by doing away with locking and synchronization.

%and propose \tuner, an automatic tuner for its hyperparameters.
%A key component of \tuner is principled momentum-tuning in both synchronous and asynchronous settings.
%\tuner is competitive or better that state-of-the-art adaptive methods and is asynchrony-aware by using a novel design based on measuring the level of momentum on the fly and using a closed-loop control module for momentum.

%\outline{[OUR RESULTS]}
%We demonstrate experimentally, in Section~\ref{sec:experiments}, that
%for a large class of networks, hand-tuned momentum SGD is competitive with Adam ($0.88-1.82\times$), even though it uses fixed hyperparameters throughout execution.
%On ResNets, momentum SGD can achieve a $1.25-1.82\times$ speedup over Adam.
%\tuner is competitive with, and sometimes better, than state-of-the-art adaptive methods. The speedup is over $2\times$ on ResNets and $1.18\times$ on LSTMs.
%Other adaptive algorithms do not tune their momentum parameter and suffer, as a result, in asynchronous settings.
%Hand-tuning momentum for Adam can improve its convergence rate by up to $2\times$, when using $16$ asynchronous workers.
%Finally, closing the momentum loop in \tuner yields a speedup of $1.3-3\times$, when using $16$ asynchronous workers.
%Closed-loop \tuner achieves a speedup of about $2.7\times$ over Adam on 16 asynchronous workers.
%\begin{itemize}
%\item For a large class of networks, hand-tuned momentum SGD is competitive with Adam ($0.88-1.82\times$), even though it uses fixed hyperparameters throughout execution.
%On ResNets, momentum SGD can achieve a $1.25-1.82\times$ speedup over Adam.
%\item \tuner is competitive with, and sometimes better, than state-of-the-art adaptive methods. The speedup is over $2\times$ on ResNets and $1.18\times$ on LSTMs.
%\item  Other adaptive algorithms do not tune their momentum parameter and suffer, as a result, in asynchronous settings.
%Hand-tuning momentum for Adam can improve its convergence rate by up to $2\times$, when using $16$ asynchronous workers.
%\item  Closing the momentum loop in \tuner yields a speedup of $1.3-3\times$, when using $16$ asynchronous workers.
%Closed loop \tuner achieves a speedup of about $2.7\times$ over Adam on 16 asynchronous workers.
%\end{itemize}

\outline{[empirical performance statement on \tuner]}
We introduce our contributions in Sections~\ref{sec:momentum_operator},\ref{sec:sync_tuner},\ref{sec:async_tuner}.
In Section~\ref{sec:experiments}, we demonstrate empirically that
\tuner offers \jianedits{competitive or significantly better performance} compared to:
(i) hand-tuned momentum SGD (a speedup of up to 2.3x);
and (ii) hand-tuned Adam (up to 2.8x speedup).
	\jianedits{In an asynchronous setting, we demonstrate that Adam, the state-of-the-art adaptive method, suffers from lack of momentum tuning.}
%Specifically, when we hand-tune Adam's momentum, its asynchronous performance improves by at least 2.5x.
In the same setting, the closed-loop control architecture speeds up \tuner by up to 2x,
%closing \tuner's momentum loop improves its performance by 1.5x-3x.
\jianedits{allowing \Asynctuner to outperform Adam with a more than 2.7x speedup}.
\jianedits{As a conclusion, we present related work in Section~\ref{sec:related} and discussion in Section~\ref{sec:discussion}.}
%We conclude with related work in Section~\ref{sec:related} and discussion in Section~\ref{sec:discussion}

%We introduce our contributions in Sections~\ref{sec:momentum_operator},\ref{sec:sync_tuner},\ref{sec:async_tuner}.
%Section~\ref{sec:experiments} gives experimental validation of our claims and we conclude with related work in Section~\ref{sec:related} and discussion in Section~\ref{sec:discussion}.


\chapter{Discussion}
The findings of the study lead to the following points of general interest, beginning with the pairwise interactions. For the $T^p - T^-$ pairwise competition, the limitation of testosterone for $T^p$ and that of oxygen for $T^-$ has a major influence on competition and its outcomes. Increasing the limitation of a given resource for one cell relative to the other leads to the more-limited cell going extinct, regardless of specific identity. Only when the limitations are balanced between two types was coexistence observed. Resource levels can therefore act as control levers of the strength of competitive interactions between cell types and therefore determine the feasibility of coexistence. In addition to resource limitations, the relative initial seeding proportion of the cells can push the outcome in favour of the dominant cell. $T^-$ in conventional terms has an advantage due to its shorter doubling time and the requirement of one less resource. A model that doesn’t account for explicit resource dynamics and limitations might therefore predict that $T^-$ always wins. Although competition coefficients can be made to disfavour $T^-$ , justification for such an artefact isn’t straightforward, particularly compared to the emergence of coexistence we observed in our model due to resource limitation effects.

Similar to $T^p - T^-$, the relative limitation for a given resource of one cell over the other, as well as the relative seeding proportion of the cells, both influence the outcomes of the $T^+ - T^p$ pairwise competition. Resource limitations can in fact be directly compared here unlike $T^p - T^-$, as both the cell types share the same qualitative resource dependencies. $T^p$ has an advantage over $T^+$ with oxygen limitations as the latter requires the former for testosterone. However, with testosterone limitation, $T^p$ has a disadvantage relative to $T^+$ as it would be doubly growth-limited by both testosterone and $T^+$ density. Even though symmetric limitation of a resource across both the cell types produces a similar effect, the difference with testosterone is much more pronounced than with oxygen limitation, possibly indicating a stronger dependence on testosterone for these two cell types.

The general trend discussed above holds for the three-way competition as well. Due to the doubling time advantage of $T^-$ and homogeneous limitations across cell types, testosterone limitations on $T^p$ and $T^+$ have a higher influence on maintaining coexistence between the cells. In this context, it has been possible to identify zones of resource limitation for both testosterone and oxygen where the system goes from coexistence to $T^-$ domination. Strong testosterone limitation without a numerical advantage of higher initial seeding density for $T^p$ leads to $T^-$ domination with no influence from oxygen. Weak testosterone limitation with a numerical advantage for $T^p$ leads to coexistence with no influence of oxygen. Meanwhile, in the other cases where testosterone limitation is intermediate, the outcomes of competition are pushed either towards coexistence or $T^-$ dominance by the oxygen limitation.

This framework used for studying competition alone then helps us understand the outcomes of therapy in mechanistic terms. $T^p$ and $T^+$ are the only cell types to depend on testosterone and hence only they respond to abiraterone. SOC creates an additional limitation of testosterone by reducing the production rates and hence pushes these two cell types to extinction. AT would have no influence where $T^p$ and $T^+$ are pushed to extinction by competition alone and maximum influence when the tumour is dominated by $T^p$ and $T^+$. The resource limitations and seeding proportions therefore have an influence on the success of AT. The success of AT also depends crucially on the therapy window. Higher window would lead to better success, but at the increased physiological cost of maintaining a larger tumour. Meanwhile, with a smaller window achieving control would be more difficult and there would be a higher risk of competitive release. While this mechanistic information makes for a thorough understanding of the system from first principles, it also highlights the gap between the modelling approach and clinical reality. It is clear that application of such ecologically-aware treatment strategies would also require a quantum shift in the nature of information that can be realistically obtained from a cancer patient over meaningful timescales.

Insofar as eliminating the treatment-resistant cell type, all the therapy strategies we tried have been a failure. But, some of this failure has been informative, and it has been possible to avoid or delay competitive release in some cases by maintaining a non-zero population of the responsive cell types $T^+$ and $T^p$. This is broadly in line with current thinking in the field regarding the goals of AT, which are more focused on achieving control than a complete cure. It is worth noting however, that the total tumour burden even when control was achieved was very close to the model’s maximum effective carrying capacity. This could point to an important gap in the conceptualisation of AT that does not include the physiological cost of control over cure. It may then be worth investigating if AT could also be designed to address ways of minimising this cost alongside tumour control.

This study has been an attempt at a proof of concept to illustrate how ecological dynamics within a cancer system can inform progression as well as therapeutic decisions and outcomes. Based on its results so far, the following lines of further development are worth highlighting:
\begin{enumerate}
  \item The exploration of combination therapy in this study has been limited. While the effects of docetaxel in the model were determined based on available experimental data, the ways in which it can be applied are still open modelling questions that can include the frequency of docetaxel, the magnitude of the dose, and the phase shift between abiraterone and docetaxel. Methods of combination therapy other than docetaxel are also known clinically (radiation, steroids, etc) and some of these can be incorporated into the framework of the current model to understand the dynamics of combination therapy better.
  \item The cell types considered here are an oversimplification of actual cells in a biological system, which are highly variable in their functions and phenotypes. Cancer systems in particular can show an even higher degree of such heterogeneity due to their higher mutation rates and genomic instability. Bringing this to bear on the modelling approach we have taken, exploring a heterogeneity of cellular response across cell types to resource availability would be of interest here due to its strong influence on the outcomes of somatic competition and its effect on therapy. This is how pairwise competition has been explored in this study, and its extension to three-way competition should be informative.
  \item In addition to heterogeneity, the cells can also switch their phenotype based on environmental conditions due to phenotypic plasticity. While some theoretical studies have explored the effect of mutational changes within the context of cell competition \cite{Snippert}, phenotypic plasticity remains understudied. It is possible, however, that implementing plasticity or heterogeneous responses would be much easier with an individual based model, which opens up whole new possibilities of exploring the ecological dynamics of the system.
  \item An individual-based model also allows for the addition of spatial heterogeneity to the system, which is an important component of biological variation in cancer populations. Solid tumours in particular have well-known gradients of resources between the tumour core and edge \cite{Fontaine}, which could again open up even more channels of investigation.
\end{enumerate}

These are possible ways in which the current study could be extended and broadened in scope. However, a more general comment on the modelling approach itself is also justified at this point.

Mathematical modelling is a really powerful tool in biology which helps in understanding a system without use of an actual system for the main experimentation. This can be very useful when experimenting on the actual system or biological model is not possible due to ethics, health risks, etc. However, one must keep in mind that all biological models are simplifications and involve many assumptions. Quoting \cite{Box} ``All models are wrong, but some are useful". There are no objective criteria for what makes a good model, or a fair set of prior assumptions. Every model makes assumptions that are manifestations of constraints inherent to the mathematical framework and data availability.

The concept of carrying capacity in a logistic or Lotka-Volterra system is a debated topic and even more so in cancer systems \cite{McLeod,Deisboeck}. Our model also assumes a carrying capacity derived from a rather arbitrary equilibrium value, $y_i^*$, given in \autoref{K_eq}. However, this arbitrariness is partially alleviated since this carrying capacity is not explicitly fixed and instead is dynamically affected by the current concentration of resources. Nevertheless, it still suffers from the same setbacks of any model that invokes a carrying capacity, which is both difficult to define biologically (especially for cancer populations characterised by ``uncontrolled" growth) and usually impossible to separate from the intrinsic growth rate. This leads to the possibility that a modelling approach that does not involve the carrying capacity at all where the resource availability is directly linked to growth rate in a consumer-resource type model may be better suited to study cancer systems that are typically at the edge of resource availability. Furthermore, comparing the differences between such alternate modelling strategies could itself form an informative study of modelling assumptions and their impact on inferences.

The inherently exponential nature of the model made it very sensitive to parameter values and to small fluctuations in environmental conditions. This is borne out particularly clearly in how this system responds to the application of therapy almost instantaneously. The rapidity of these responses could likely hide subtler dynamics that may be more accessible to a different modelling approach that is designed to pick up on them.

Our modelling approach has been more mechanistic in nature than data driven. We have started with parameters from fundamental processes governing the system for the overall behaviour to emerge out of it. A data driven model on the other hand, fits the parameters to clinical data. One could argue that such a data driven model reflects closer to reality since it follows the same dynamics, however, such models don’t give valuable insight into these fundamental processes and act as black boxes. It may then be useful to explore ways of integrating clinical data more closely into mechanistic models, potentially resulting in mechanistic insight that can be applied more directly to clinical practice.