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\documentstyle[11pt,reduce]{article} \title{GROEBNER: A Package for Calculating Gr\"obner Bases} \date{} \author{ H. Melenk \& W. Neun \\[0.05in] Konrad--Zuse--Zentrum \\ f\"ur Informationstechnik Berlin \\ Heilbronner Strasse 10 \\ D--10711 Berlin-Wilmersdorf \\ Germany \\[0.05in] Email: melenk@sc.zib--berlin.de \\[0.05in] and \\[0.05in] H.M. M\"oller \\[0.05in] Fernuniversit\"at Hagen \\ FB Math und Informatik\\ Postfach 940 \\ D--58084 Hagen \\ Germany\\[0.05in] Email: Michael.Moeller@fernuni-hagen.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 the previous version, written by R. Gebauer, A.C. Hearn, H. Kredel and 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}. \section{Compatibility} Important modifications of the present GROEBNER package compared with the REDUCE 3.5 GROEBNER package: \begin{itemize} \item The variables of the polynomial ring are now declared in the $TORDER$ statement together with the term order mode. The old style (including the variable list in the function calls) is still accepted, but will not be supported in future versions. \item The term order mode $private$ has been eliminated. Instead the mode $matrix$ has been introduced, which allows you to define any compatible term order. For optimal efficiency a matrix can be compiled. \item Polynomial side relations in the parameter domain are now considered inside the Gr\"obner calculations. You can enter them as ordinary $LET$ rules. \item There is an extension $NCPOLY$ for computing with non--commutative ideals of solvable type. $NCPOLY$ is loaded as a separate module, but it uses internally the functions of $GROEBNER$. \end{itemize} \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. 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 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). 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. \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. \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}. % Chapter 2 \section{Loading of the Package} The following command loads the package into REDUCE (this syntax may vary according to implementation): \begin{center} load groebner; \end{center} The package contains various operators, and switches for control over the reduction process. These are discussed in the following. % Chapter 3 \section{The Basic Operators} % Section 3.1 \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, REV\-GRAD\-LEX (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 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 zeros 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 ``Kredel-Weispfenning" algorithm is used (see \cite{Kredel:88a}, extended to general ordering in \cite{BeWei:93}. \subsection{GLEXCONVERT: Conversion of an Arbitrary Gr\"obner Basis 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. \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} \subsection{GROEBNERF: Factorizing Gr\"obner Bases} % Subsection 3.4.1 \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. % Subsection 3.4.2 \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 \\ TRGROEB1 \end{tabular} \end{center} \subsubsection*{Additional switches for GROEBNERF:} \begin{description} \ttindex{GROEBRES} \item[GROEBRES] -- If ON, a resultant is calculated under certain circumstances (one bivariate $H$--polynomial is followed by another one). This sometimes shortens the calculation. By default GROEBRES is off. \ttindex{TRGROEBR} \item[TRGROEBR] -- All intermediate partial basis are printed when detected. By default TRGROEBR is off. \end{description} {\bf 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. \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 GROEBRESTRICTION:=NONEGATIVE;} \\ \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 $. % Subsection 3.5.2 \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 $ exp $ 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{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. \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, $C_{ij}$ 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. % *** SO BESSER ?? \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 if your REDUCE is equipped with a LISP compiler; 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. % Section 4.1 \subsection{Solutions Based on Lex Type Gr\"obner Bases} % Subsection 4.1.1 \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} \[ GROESOLVE(\{x**2 - y**2 - a, p*x+q*y+s\},\{x,y\}); \] %{\small \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} %} % Subsection 4.1.2 \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} % Subsection 4.1.3 \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. \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 $\{exp1, exp2, \ldots , expm\}$ is a list of any number of expressions or equations. HILBERTPOLYNOMIAL calculates the Hilbert polynomial of the ideal with basis $\{exp1, exp2, \ldots , expm\}$ 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 $\{exp1, exp2,\ldots$, $ 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 $\{exp1, exp2, \ldots , expm\}$: 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 $\{var1, var2, \ldots , varn\}$ 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. \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 large bases. % Chapter 5. \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 5.1 \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 5.2 \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}