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51.1 Introduction to grobner | ||
51.2 Functions and Variables for grobner |
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grobner
is a package for working with Groebner bases in Maxima.
A tutorial on Groebner Bases can be found at
http://www.geocities.com/CapeCanaveral/Hall/3131/
To use the following functions you must load the `grobner.lisp' package.
load(grobner);
A demo can be started by
demo("grobner.demo");
or
batch("grobner.demo")
Some of the calculation in the demo will take a lot of time therefore the output `grobner-demo.output' of the demo can be found in the same directory as the demo file.
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The package was written by Marek Rychlik http://alamos.math.arizona.edu and is released 2002-05-24 under the terms of the General Public License(GPL) (see file `grobner.lisp'. This documentation was extracted from the files
`README', `grobner.lisp', `grobner.demo', `grobner-demo.output'
by Günter Nowak. Suggestions for improvement of the documentation can be discussed at the maxima-mailing-list maxima@math.utexas.edu. The code is a little bit out of date now. Modern implementation use the fast F4 algorithm described in "A new efficient algorithm for computing Gröbner bases (F4)", Jean-Charles Faugère, LIP6/CNRS Université Paris VI, January 20, 1999.
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lex
pure lexicographic,
default order for monomial comparisons
grlex
total degree order, ties broken by lexicographic
grevlex
total degree, ties broken by reverse lexicographic
invlex
inverse lexicographic order
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Default value: lex
This global switch controls which monomial order is used in polynomial and
Groebner Bases calculations. If not set, lex
will be used.
Default value: expression_ring
This switch indicates the coefficient ring of the polynomials that will be used
in grobner calculations. If not set, maxima's general expression ring
will be used. This variable may be set to ring_of_integers
if desired.
Default value: false
Name of the default order for eliminated variables in elimination-based
functions. If not set, lex
will be used.
Default value: false
Name of the default order for kept variables in elimination-based functions.
If not set, lex
will be used.
Default value: false
Name of the default elimination order used in elimination calculations. If set,
it overrides the settings in variables poly_primary_elimination_order
and
poly_secondary_elimination_order
. The user must ensure that this is a
true elimination order valid for the number of eliminated variables.
Default value: false
If set to true
, all functions in this package will return each polynomial
as a list of terms in the current monomial order rather than a maxima
general expression.
Default value: false
If set to true
, produce debugging and tracing output.
Default value: buchberger
Possible values:
buchberger
parallel_buchberger
gebauer_moeller
The name of the algorithm used to find the Groebner Bases.
Default value: false
If not false
, use top reduction only whenever possible. Top reduction
means that division algorithm stops after the first reduction.
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poly_add
, poly_subtract
, poly_multiply
and poly_expt
are the arithmetical operations on polynomials. These are performed using the
internal representation, but the results are converted back to the maxima
general form.
Adds two polynomials poly1 and poly2.
(%i1) poly_add(z+x^2*y,x-z,[x,y,z]); 2 (%o1) x y + x
Subtracts a polynomial poly2 from poly1.
(%i1) poly_subtract(z+x^2*y,x-z,[x,y,z]); 2 (%o1) 2 z + x y - x
Returns the product of polynomials poly1 and poly2.
(%i1) poly_multiply(z+x^2*y,x-z,[x,y,z])-(z+x^2*y)*(x-z),expand; (%o1) 0
Returns the syzygy polynomial (S-polynomial) of two polynomials poly1 and poly2.
Returns the polynomial poly divided by the GCD of its coefficients.
(%i1) poly_primitive_part(35*y+21*x,[x,y]); (%o1) 5 y + 3 x
Returns the polynomial poly divided by the leading coefficient. It assumes that the division is possible, which may not always be the case in rings which are not fields.
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This function parses polynomials to internal form and back. It is equivalent
to expand(poly)
if poly parses correctly to a polynomial.
If the representation is not compatible with a polynomial in variables
varlist, the result is an error. It can be used to test whether an
expression correctly parses to the internal representation. The following
examples illustrate that indexed and transcendental function variables are
allowed.
(%i1) poly_expand((x-y)*(y+x),[x,y]); 2 2 (%o1) x - y (%i2) poly_expand((y+x)^2,[x,y]); 2 2 (%o2) y + 2 x y + x (%i3) poly_expand((y+x)^5,[x,y]); 5 4 2 3 3 2 4 5 (%o3) y + 5 x y + 10 x y + 10 x y + 5 x y + x (%i4) poly_expand(-1-x*exp(y)+x^2/sqrt(y),[x]); 2 y x (%o4) - x %e + ------- - 1 sqrt(y) (%i5) poly_expand(-1-sin(x)^2+sin(x),[sin(x)]); 2 (%o5) - sin (x) + sin(x) - 1
exponentitates poly by a positive integer number. If number is not a positive integer number an error will be raised.
(%i1) poly_expt(x-y,3,[x,y])-(x-y)^3,expand; (%o1) 0
poly_content
extracts the GCD of its coefficients
(%i1) poly_content(35*y+21*x,[x,y]); (%o1) 7
Pseudo-divide a polynomial poly by the list of n polynomials polylist. Return multiple values. The first value is a list of quotients a. The second value is the remainder r. The third argument is a scalar coefficient c, such that c*poly can be divided by polylist within the ring of coefficients, which is not necessarily a field. Finally, the fourth value is an integer count of the number of reductions performed. The resulting objects satisfy the equation:
c*poly=sum(a[i]*polylist[i],i=1...n)+r.
Divide a polynomial poly1 by another polynomial poly2. Assumes that exact division with no remainder is possible. Returns the quotient.
poly_normal_form
finds the normal form of a polynomial poly with
respect to a set of polynomials polylist.
Returns true
if polylist is a Groebner basis with respect to the
current term order, by using the Buchberger criterion: for every two polynomials
h1 and h2 in polylist the S-polynomial S(h1,h2)
reduces to 0 modulo polylist.
poly_buchberger
performs the Buchberger algorithm on a list of
polynomials and returns the resulting Groebner basis.
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The k-th elimination Ideal I_k of an Ideal I over
K[ x[1],...,x[n] ] is the ideal
intersect(I, K[ x[k+1],...,x[n] ]).
The colon ideal I:J is the ideal
{h|for all w in J: w*h in I}.
The ideal I:p^inf is the ideal
{h| there is a n in N: p^n*h in I}.
The ideal I:J^inf is the ideal
{h| there is a n in N and a p in J: p^n*h in I}.
The radical ideal sqrt(I) is the ideal
{h| there is a n in N : h^n in I }.
poly_reduction
reduces a list of polynomials polylist, so that
each polynomial is fully reduced with respect to the other polynomials.
Returns a sublist of the polynomial list polylist spanning the same monomial ideal as polylist but minimal, i.e. no leading monomial of a polynomial in the sublist divides the leading monomial of another polynomial.
poly_normalize_list
applies poly_normalize
to each polynomial in
the list. That means it divides every polynomial in a list polylist by
its leading coefficient.
Returns a Groebner basis of the ideal span by the polynomials polylist. Affected by the global flags.
Returns a reduced Groebner basis of the ideal span by the polynomials polylist. Affected by the global flags.
poly_depends
tests whether a polynomial depends on a variable var.
poly_elimination_ideal
returns the grobner basis of the number-th
elimination ideal of an ideal specified as a list of generating polynomials
(not necessarily Groebner basis).
Returns the reduced Groebner basis of the colon ideal
I(polylist1):I(polylist2)
where polylist1 and polylist2 are two lists of polynomials.
poly_ideal_intersection
returns the intersection of two ideals.
Returns the lowest common multiple of poly1 and poly2.
Returns the greatest common divisor of poly1 and poly2.
See also ezgcd
, gcd
, gcdex
, and
gcdivide
.
Example:
(%i1) p1:6*x^3+19*x^2+19*x+6; 3 2 (%o1) 6 x + 19 x + 19 x + 6 (%i2) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x; 5 4 3 2 (%o2) 6 x + 13 x + 12 x + 13 x + 6 x (%i3) poly_gcd(p1, p2, [x]); 2 (%o3) 6 x + 13 x + 6
poly_grobner_equal
tests whether two Groebner Bases generate the same
ideal. Returns true
if two lists of polynomials polylist1 and
polylist2, assumed to be Groebner Bases, generate the same ideal, and
false
otherwise. This is equivalent to checking that every polynomial
of the first basis reduces to 0 modulo the second basis and vice versa. Note
that in the example below the first list is not a Groebner basis, and thus the
result is false
.
(%i1) poly_grobner_equal([y+x,x-y],[x,y],[x,y]); (%o1) false
poly_grobner_subsetp
tests whether an ideal generated by polylist1
is contained in the ideal generated by polylist2. For this test to always
succeed, polylist2 must be a Groebner basis.
Returns true
if a polynomial poly belongs to the ideal generated by
the polynomial list polylist, which is assumed to be a Groebner basis.
Returns false
otherwise.
poly_grobner_member
tests whether a polynomial belongs to an ideal
generated by a list of polynomials, which is assumed to be a Groebner basis.
Equivalent to normal_form
being 0.
Returns the reduced Groebner basis of the saturation of the ideal
I(polylist):poly^inf
Geometrically, over an algebraically closed field, this is the set of polynomials in the ideal generated by polylist which do not identically vanish on the variety of poly.
Returns the reduced Groebner basis of the saturation of the ideal
I(polylist1):I(polylist2)^inf
Geometrically, over an algebraically closed field, this is the set of polynomials in the ideal generated by polylist1 which do not identically vanish on the variety of polylist2.
polylist2 ist a list of n polynomials [poly1,...,polyn]
.
Returns the reduced Groebner basis of the ideal
I(polylist):poly1^inf:...:polyn^inf
obtained by a sequence of successive saturations in the polynomials of the polynomial list polylist2 of the ideal generated by the polynomial list polylist1.
polylistlist is a list of n list of polynomials
[polylist1,...,polylistn]
. Returns the reduced Groebner basis of the
saturation of the ideal
I(polylist):I(polylist_1)^inf:...:I(polylist_n)^inf
poly_saturation_extension
implements the famous Rabinowitz trick.
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