Matrix Operations
GROEBNER BASES _ _ _ _ _ _ _ _ _ _ _ _ introduction
The GROEBNER package calculates Groebner bases using the Buchberger algorithm and provides related algorithms for arithmetic with ideal bases, such as ideal quotients, Hilbert polynomials ( Hollmann algorithm), basis conversion ( Faugere-Gianni-Lazard-Mora algorithm), independent variable set ( Kredel-Weispfenning algorithm).
Some routines of the Groebner package are used by solve - in that context the package is loaded automatically. However, if you want to use the package by explicit calls you must load it by
load_package groebner;
For the common parameter setting of most operators in this package see ideal parameters.
IDEAL PARAMETERS
Most operators of the Groebner package compute expressions in a polynomial ring which given as <R>[<var>,<var>,...] where <R> is the current REDUCE coefficient domain. All algebraically exact domains of REDUCE are supported. The package can operate over rings and fields. The operation mode is distinguished automatically. In general the ring mode is a bit faster than the field mode. The factoring variant can be applied only over domains which allow you factoring of multivariate polynomials.
The variable sequence <var> is either declared explicitly as argument in form of a list in torder, or it is extracted automatically from the expressions. In the second case the current REDUCE system order is used (see korder) for arranging the variables. If some kernels should play the role of formal parameters (the ground domain <R> then is the polynomial ring over these), the variable sequences must be given explicitly.
All REDUCE kernels can be used as variables. But please note, that all variables are considered as independent. E.g. when using sin(a) and cos(a) as variables, the basic relation sin(a)^2+cos(a)^2-1=0 must be explicitly added to an equation set because the Groebner operators don't include such knowledge automatically.
The terms (monomials) in polynomials are arranged according to the current term order. Note that the algebraic propertie s of the computed results only are valid as long as neither the ordering nor the variable sequence changes.
The input expressions <exp> can be polynomials <p>, rational functions <n>/<d> or equations <lh>=<rh> built from polynomials or rational functions. Apart from the tracing algorithms groebnert and preducet, where the equations have a specific meaning, equations are converted to simple expressions by taking the difference of the left-hand and right-hand sides <lh>-<rh>=><p>. Rational functions are converted to polynomials by converting the expression to a common denominator form first, and then using the numerator only <n>=><p>. So eventual zeros of the denominators are ignored.
A basis on input or output of an algorithm is coded as list of expressions {<exp>,<exp>,...} .
TERM ORDER _ _ _ _ _ _ _ _ _ _ _ _ introduction
For all Groebner operations the polynomials are represented in distributive form: a sum of terms (monomials). The terms are ordered corresponding to the actual term order which is set by the torder operator, and to the actual variable sequence which is either given as explicit parameter or by the system kernel order.
TORDER _ _ _ _ _ _ _ _ _ _ _ _ operator
The operator torder sets the actual variable sequence and term order.
1. simple term order:
torder(<vl>, <m>)
where <vl> is a list of variables ( kernels) and <m> is the name of a simple term order mode lex term order, gradlex term order, revgradlex term order or another implemented parameterless mode.
2. stepped term order:
torder(<vl>,<m>,<n>)
where <m> is the name of a two step term order, one of gradlexgradlex term order, gradlexrevgradlex term order, lexgradlex term order or lexrevgradlex term order, and <n> is a positive integer.
3. weighted term order
torder(<vl>, weighted, <n>,<n>,...);
where the <n> are positive integers, see weighted term order.
4. matrix term order
torder(<vl>, matrix, <m>);
where <m> is a matrix with integer elements, see torder_compile.
5. compiled term order
torder(<vl>, co);
where <co> is the name of a routine generated by torder_compile.
tordersets the variable sequence and the term order mode. If the an empty list is used as variable sequence, the automatic variable extraction is activated. The defaults are the empty variable list an the lex term order. The previous setting is returned as a list.
Alternatively to the above syntax the arguments of torder may be collected in a list and passed as one argument to torder.
TORDER_COMPILE _ _ _ _ _ _ _ _ _ _ _ _ operator
A matrix can be converted into a compilable LISP program for faster execution by using
torder_compile(<name>,<mat>)
where <name> is an identifier for the new term order and <mat> is an integer matrix to be used as matrix term order. Afterwards the term order can be activated by using <name> in a torder expression. The resulting program is compiled if the switch comp is on, or if the torder_compile expression is part of a compiled module.
LEX TERM ORDER
The terms are ordered lexicographically: two terms t1 t2 are compared for their degrees along the fixed variable sequence: t1 is higher than t2 if the first different degree is higher in t1. This order has the elimination property for groebner basis calculations. If the ideal has a univariate polynomial in the last variable the groebner basis will contain such polynomial. Lex is best suited for solving of polynomial equation systems.
GRADLEX TERM ORDER
The terms are ordered first with their total degree, and if the total degree is identical the comparison is lex term order. With groebner basis calculations this term order produces polynomials of lowest degree.
REVGRADLEX TERM ORDER
The terms are ordered first with their total degree (degree sum), and if the total degree is identical the comparison is the inverse of lex term order. With groebner and groebnerf calculations this term order is similar to gradlex term order; it is known as most efficient ordering with respect to computing time.
GRADLEXGRADLEX TERM ORDER
The terms are separated into two groups where the second parameter of the torder call determines the length of the first group. For a comparison first the total degrees of both variable groups are compared. If both are equal gradlex term order comparison is applied to t he first group, and if that does not decide gradlex term order is applied for the second group. This order has the elimination property for the variable groups. It can be used e.g. for separating variables from parameters.
GRADLEXREVGRADLEX TERM ORDER
Similar to gradlexgradlex term order, but using revgradlex term order for the second group.
LEXGRADLEX TERM ORDER
Similar to gradlexgradlex term order, but using lex term order for the first group.
LEXREVGRADLEX TERM ORDER
Similar to gradlexgradlex term order, but using lex term order for the first group revgradlex term order for the second group.
WEIGHTED TERM ORDER
establishes a graduated ordering similar to gradlex term order, where the exponents first are multiplied by the given weights. If there are less weight values than variables, the weight list is extended by ones. If the weighted degree comparison is not decidable, the lex term order is used.
GRADED TERM ORDER
establishes a cascaded term ordering: first a graduated ordering similar to gradlex term order is used, where the exponen ts first are multiplied by the given weights. If there are less weight values than variables, the weight list is extended by ones. If the weighted degree comparison is not decidable, the term ordering described in the following parameters of the torder command is used.
MATRIX TERM ORDER
Any arbitrary term order mode can be installed by a matrix with integer elements where the row length corresponds to the variable number. The matrix must have at least as many rows as columns. It must have full rank, and the top nonzero element of each column must be positive.
The matrix term order mode defines a term order where the exponent vectors of the monomials are first multiplied by the matrix and the resulting vectors are compared lexicographically.
If the switch comp is on, the matrix is converted into a compiled LISP program for faster execution. A matrix can also be compiled explicitly, see torder_compile.
Term order
GVARS _ _ _ _ _ _ _ _ _ _ _ _ operator
gvars({<exp>,<exp>,... })
where <exp> are expressions or equations.
gvarsextracts from the expressions the kernels which can play the role of variables for a groebner or groebnerf calculation.
GROEBNER _ _ _ _ _ _ _ _ _ _ _ _ operator
where {exp, ... } is a list of expressions or equations.
The operator groebner implements the Buchberger algorithm for computing Groebner bases for a given set of expressions with respect to the given set of variables in the order given. As a side effect, the sequence of variables is stored as a REDUCE list in the shared variable gvarslast - this is important in cases where the algorithm rearranges the variable sequence because groebopt is on.
groebner({x**2+y**2-1,x-y})
{X - Y,2*Y**2 -1}
_ _ _ groebnerfoperator
_ _ _ gvarslast variable
_ _ _ groebopt switch
_ _ _ groebprereduce switch
_ _ _ groebfullreduction switch
_ _ _ gltbasis switch
_ _ _ gltb variable
_ _ _ glterms variable
_ _ _ groebstat switch
_ _ _ trgroeb switch
_ _ _ trgroebs switch
_ _ _ groebprot switch
_ _ _ groebprotfile variable
_ _ _ groebnert operator
GROEBNER\_WALK _ _ _ _ _ _ _ _ _ _ _ _ operator
The operator groebner_walk computes a lex basis from a given graded (or weighted) one.
groebner_walk(<g>)
where <g> is a graded basis (or weighted basis with a weight vector with one repeated element) of the polynomial ideal. Groebner_walk computes a sequence of monomial bases, each time lifting the full system to a complete basis. Groebner_walk should be called only in cases, where a normal kex computation would take too much computer time.
The operator torder has to be called before in order to define the variable sequence and the term order mode of <g>.
The variable gvarslast is not set.
Do not call groebner_walk with on groebopt.
Groebner_walkincludes some overhead (such as e. g. computation with division). On the other hand, sometimes groebner_walk is faster than a direct lex computation.
GROEBOPT _ _ _ _ _ _ _ _ _ _ _ _ switch
If groebopt is set ON, the sequence of variables is optimized with respect to execution speed of groebner calculations; note that the final list of variables is available in gvarslast. By default groebopt is off, conserving the original variable sequence.
An explicitly declared dependency using the depend declaration supersedes the variable optimization.
guarantees that a will be placed in front of x and y.
GVARSLAST _ _ _ _ _ _ _ _ _ _ _ _ variable
After a groebner or groebnerf calculation the actual variable sequence is stored in the variable gvarslast. If groebopt is on gvarslast shows the variable sequence after reordering.
GROEBPREREDUCE _ _ _ _ _ _ _ _ _ _ _ _ switch
If groebprereduce set ON, groebner and groebnerf try to simplify the input expressions: if the head term of an input expression is a multiple of the head term of another expression, it can be reduced; these reductions are done cyclicly as long as possible in order to shorten the main part of the algorithm.
By default groebprereduce is off.
GROEBFULLREDUCTION _ _ _ _ _ _ _ _ _ _ _ _ switch
If groebfullreduction set off, the polynomial reduction steps during groebner and groebnerf are limited to the pure head term reduction; subsequent terms are reduced otherwise.
By default groebfullreduction is on.
GLTBASIS _ _ _ _ _ _ _ _ _ _ _ _ switch
If gltbasis set on, the leading terms of the result basis of a groebner or groebnerf calculation are extracted. They are collected as a basis of monomials, which is available as value of the global variable gltb.
GLTB _ _ _ _ _ _ _ _ _ _ _ _ variable
See gltbasis
GLTERMS _ _ _ _ _ _ _ _ _ _ _ _ variable
If the expressions in a groebner or groebnerf call contain parameters (symbols which are not member of the variable list), the share variable glterms is set to a list of expression which during the calculation were assumed to be nonzero. The calculated bases are valid only under the assumption that all these expressions do not vanish.
GROEBSTAT _ _ _ _ _ _ _ _ _ _ _ _ switch
if groebstat is on, a summary of the groebner or groebnerf computation is printed at the end including the computing time, the number of intermediate H polynomials and the counters for the criteria hits.
TRGROEB _ _ _ _ _ _ _ _ _ _ _ _ switch
if trgroeb is on, intermediate H polynomials are printed during a groebner or groebnerf calculation.
TRGROEBS _ _ _ _ _ _ _ _ _ _ _ _ switch
if trgroebs is on, intermediate H and S polynomials are printed during a groebner or groebnerf calculation.
GZERODIM? _ _ _ _ _ _ _ _ _ _ _ _ operator
gzerodim!?(<basis>)
where <bas> is a Groebner basis in the current term order with the actual setting (see ideal parameters).
gzerodim!?tests whether the ideal spanned by the given basis has dimension zero. If yes, the number of zeros is returned, nil otherwise.
GDIMENSION _ _ _ _ _ _ _ _ _ _ _ _ operator
where <bas> is a groebner basis in the current term order (see ideal parameters). gdimension computes the dimension of the ideal spanned by the given basis and returns the dimension as an integer number. The Kredel-Weispfenning algorithm is used: the dimension is the length of the longest independent variable set, see gindependent_sets
GINDEPENDENT\_SETS _ _ _ _ _ _ _ _ _ _ _ _ operator
where <bas> is a groebner basis in any term order (which must be the current term order) with the specified variables (see ideal parameters).
Gindependent_setscomputes 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. The Kredel-Weispfenning algorithm is used.
DD_GROEBNER _ _ _ _ _ _ _ _ _ _ _ _ operator
For a homogeneous system of polynomials under graded term order, gradlex term order, revgradlex term order
or weighted term order a Groebner Base can be computed with limiting the grade of the intermediate S polynomials:
dd_groebner(<d1>,<d2>,<plist>)
where <d1> is a non negative integer and <d2> is an integer or ``infinity". A pair of polynomials is considered only if the grade of the lcm of their head terms is between <d1> and <d2>. For the term orders graded or weighted the (first) weight vector is used for the grade computation. Otherwise the total degree of a term is used.
GLEXCONVERT _ _ _ _ _ _ _ _ _ _ _ _ operator
where <bas> is a groebner basis in the current term order, <mx> (optional) is a positive integer and <nvl> (optional) is a list of variables (see ideal parameters).
The operator glexconvert converts the basis of a zero-dimensional ideal (finite number of isolated solutions) from arbitrary ordering into a basis under lex term order.
The parameter <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.
If <newvars> is a list with one element, the minimal univariate polynomial is computed.
<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.
During the call the term order of the input basis must be active.
GREDUCE _ _ _ _ _ _ _ _ _ _ _ _ operator
greduce(exp, {exp1, exp2, ... , expm})
where exp is an expression, and {exp1, exp2, ... , expm} is a list of expressions or equations.
greduceis functionally equivalent with a call to groebner and then a call to preduce.
PREDUCE _ _ _ _ _ _ _ _ _ _ _ _ operator
preduce(<p>, {<exp>, ... })
where <p> is an expression, and {<exp>, ... } is a list of expressions or equations.
Preducecomputes the remainder of exp modulo the given set of polynomials resp. equations. This result is unique (canonical) only if the given set is a groebner basis under the current term order
see also: preducet operator.
IDEALQUOTIENT _ _ _ _ _ _ _ _ _ _ _ _ operator
idealquotient({<exp>, ...}, <d>)
where {<exp>,...} is a list of expressions or equations, <d> is a single expression or equation.
Idealquotientcomputes the ideal quotient: ideal spanned by the expressions {<exp>,...} divided by the single polynomial/expression <f>. The result is the groebner basis of the quotient ideal.
HILBERTPOLYNOMIAL _ _ _ _ _ _ _ _ _ _ _ _ operator
where <bas> is a groebner basis in the current term order.
The degree of the Hilbert polynomial is the dimension of the ideal spanned by the basis. For an ideal of dimension zero the Hilbert polynomial is a constant which is the number of common zeros of the ideal (including eventual multiplicities). The Hollmann algorithm is used.
SATURATION _ _ _ _ _ _ _ _ _ _ _ _ operator
saturation({<exp>, ...}, <p>)
where {<exp>,...} is a list of expressions or equations, <p> is a single polynomial.
Saturationcomputes the quotient of the polynomial <p> and a power (with unknown but finite exponent) of the ideal built from {<exp>, ...}. The result is the computed quotient. Saturation calls idealquotient several times until the result does not change any more.
Basic Groebner operators
GROEBNERF _ _ _ _ _ _ _ _ _ _ _ _ operator
groebnerf({<exp>, ...}[,{},{<nz>, ... }]);
where {<exp>, ... } is a list of expressions or equations, and {<nz>,... } is an optional list of polynomials to be considered as non zero for this calculation. An empty list must be passed as second argument if the non-zero list is specified.
groebnerftries to separate polynomials into individual factors and to branch the computation in a recursive manner (factorization tree). The result is a list of partial Groebner bases. 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 third parameter of groebnerf declares some polynomials nonzero. If any of these is found in a branch of the calculation the branch is canceled.
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,x});
{{Y - 3,X},
2
{2*Y + 2*X - 1,2*X - 5*X - 5}}
_ _ _ groebresmaxvariable
_ _ _ groebmonfac variable
_ _ _ groebrestriction variable
_ _ _ groebner operator
_ _ _ gvarslast variable
_ _ _ groebopt switch
_ _ _ groebprereduce switch
_ _ _ groebfullreduction switch
_ _ _ gltbasis switch
_ _ _ gltb variable
_ _ _ glterms variable
_ _ _ groebstat switch
_ _ _ trgroeb switch
_ _ _ trgroebs switch
_ _ _ groebnert operator
GROEBMONFAC _ _ _ _ _ _ _ _ _ _ _ _ variable
The variable groebmonfac is connected to the handling of monomial factors. A monomial factor is a product of variable powers 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 groebnerf the multiplicity of monomial factor s is lowered to the value of the shared variable groebmonfac which by default is 1 (= monomial factors remain present, but their multiplicity is brought down). With groebmonfac:= 0 the monomial factors are suppressed completely.
GROEBRESMAX _ _ _ _ _ _ _ _ _ _ _ _ variable
The variable groebresmax controls during groebnerf calculations the number of partial results. Its default value is 300. If more partial results are calculated, the calculation is terminated.
GROEBRESTRICTION _ _ _ _ _ _ _ _ _ _ _ _ variable
During groebnerf calculations irrelevant branches can be excluded by setting the variable groebrestriction. The following restrictions are implemented:
groebrestriction:= nonnegative
groebrestriction:= positive
groebrestriction:= zeropoint
With nonnegative branches are excluded where one polynomial has no nonnegative real zeros; with positive the restriction is sharpened to positive zeros only. The restriction zeropoint excludes all branches which do not have the origin (0,0,...0) in their solution set.
Factorizing Groebner bases
GROEBPROT _ _ _ _ _ _ _ _ _ _ _ _ switch
If groebprot is ON the computation steps during preduce, greduce and groebner are collected in a list which is assigned to the variable groebprotfile.
GROEBPROTFILE _ _ _ _ _ _ _ _ _ _ _ _ variable
See groebprot switch.
GROEBNERT _ _ _ _ _ _ _ _ _ _ _ _ operator
groebnert({<v>=<exp>,...})
where <v> are kernels (simple or indexed variables ), <exp> are polynomials.
groebnertis functionally equivalent to a groebner call for {<exp>,...}, but the result is a set of equations where the left-hand sides are the basis elements while the right-hand sides are the same values expressed as combinations of the input formulas, expressed in terms of the names <v>
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}
PREDUCET _ _ _ _ _ _ _ _ _ _ _ _ operator
preduce(<p>,{<v>=<exp>...})
where <p> is an expression, <v> are kernels (simple or indexed variables), exp are polynomials.
preducetcomputes the remainder of <p> modulo {<exp>,...} similar to preduce, but the result is an equation which expresses the remainder as combination of the polynomials.
GB2 := {G1=2*X - Y + 1,G2=9*Y**2 - 2*Y - 199}
preducet(q=x**2,gb2);
- 16*Y + 208= - 18*X*G1 - 9*Y*G1 + 36*Q + 9*G1 - G2