@menu
* Introduction to grobner ::
* Functions and Variables for grobner ::
@end menu

@node Introduction to grobner, Functions and Variables for grobner, Top, Top
@section Introduction to grobner

@code{grobner} is a package for working with Groebner bases in Maxima.

@noindent
To use the following functions you must load the @file{grobner.lisp} package.

@example
load(grobner);
@end example

@noindent
A demo can be started by 
@example
demo("grobner.demo");
@end example

@noindent
or
@example
batch("grobner.demo")
@end example

@noindent
Some of the calculation in the demo will take a lot of time 
therefore the output @file{grobner-demo.output} of the demo can 
be found in the same directory as the demo file.

@subsection Notes on the grobner package
The package was written by 

@noindent
Marek Rychlik 

@noindent
@url{http://alamos.math.arizona.edu}

@noindent
and is released 2002-05-24 under the terms of the General Public License(GPL) (see file @file{grobner.lisp}).
This documentation was extracted from the files
@flushleft
@file{README}, @file{grobner.lisp}, @file{grobner.demo}, @file{grobner-demo.output}
@end flushleft

@noindent
by G@"unter Nowak. Suggestions for improvement of the documentation can 
be discussed at the @emph{maxima}-mailing-list @email{maxima@@math.utexas.edu}.
The code is a little bit out of date now. Modern implementation use the fast @emph{F4} algorithm described in 
@smallformat
A new efficient algorithm for computing Gr@"obner bases (F4) 
Jean-Charles Faug@`ere
LIP6/CNRS Universit@'e Paris VI 
January 20, 1999
@end smallformat

@opencatbox
@category{Groebner bases} @category{Share packages} @category{Package grobner}
@closecatbox

@subsection Implementations of admissible monomial orders in grobner

@itemize @bullet
@item @code{lex}

pure lexicographic,
default order for monomial comparisons
@item @code{grlex}

total degree order, ties broken by lexicographic

@item @code{grevlex}

total degree, ties broken by reverse lexicographic

@item @code{invlex}

inverse lexicographic order

@end itemize

@node Functions and Variables for grobner, , Introduction to grobner, Top
@section Functions and Variables for grobner 

@subsection Global switches for grobner

@defvr {Option variable} poly_monomial_order
Default value: @code{lex}

This global switch controls which monomial order is used in polynomial and Groebner Bases calculations. If not set, @code{lex} will be used.

@opencatbox
@category{Package grobner}
@closecatbox

@end defvr


@defvr {Option variable} poly_coefficient_ring
Default value: @code{expression_ring}

This switch indicates the coefficient ring of the polynomials that
will be used in grobner calculations. If not set, @emph{maxima's} general
expression ring will be used. This variable may be set to
@code{ring_of_integers} if desired.

@opencatbox
@category{Package grobner}
@closecatbox

@end defvr

@defvr {Option variable} poly_primary_elimination_order
Default value: @code{false}

Name of the default order for eliminated variables in
elimination-based functions. If not set, @code{lex} will be used.

@opencatbox
@category{Package grobner}
@closecatbox

@end defvr

@defvr {Option variable} poly_secondary_elimination_order
Default value: @code{false}

Name of the default order for kept variables in elimination-based functions. If not set, @code{lex} will be used.

@opencatbox
@category{Package grobner}
@closecatbox

@end defvr

@defvr {Option variable} poly_elimination_order
Default value: @code{false}

Name of the default elimination order used in elimination
calculations. If set, it overrides the settings in variables
@code{poly_primary_elimination_order} and @code{poly_secondary_elimination_order}.
The user must ensure that this is a true elimination order valid
for the number of eliminated variables. 

@opencatbox
@category{Package grobner}
@closecatbox

@end defvr

@defvr {Option variable} poly_return_term_list
Default value: @code{false}

If set to @code{true}, all functions in this package will return each
polynomial as a list of terms in the current monomial order rather
than a @emph{maxima} general expression.

@opencatbox
@category{Package grobner}
@closecatbox

@end defvr

@defvr {Option variable} poly_grobner_debug
Default value: @code{false}

If set to @code{true}, produce debugging and tracing output.

@opencatbox
@category{Package grobner}
@closecatbox

@end defvr

@defvr {Option variable} poly_grobner_algorithm
Default value: @code{buchberger}

Possible values: 
@itemize
@item @code{buchberger}
@item @code{parallel_buchberger}
@item @code{gebauer_moeller}
@end itemize

The name of the algorithm used to find the Groebner Bases.

@opencatbox
@category{Package grobner}
@closecatbox

@end defvr

@defvr {Option variable} poly_top_reduction_only
Default value: @code{false}

If not @code{false}, use top reduction only whenever possible. Top
reduction means that division algorithm stops after the first
reduction.

@opencatbox
@category{Package grobner}
@closecatbox

@end defvr

@subsection Simple operators in grobner
@code{poly_add}, @code{poly_subtract}, @code{poly_multiply} and @code{poly_expt}
are the arithmetical operations on polynomials.
These are performed using the internal representation, but the results are converted back to the
@emph{maxima} general form.

@deffn {Function} poly_add (@var{poly1}, @var{poly2}, @var{varlist})
Adds two polynomials @var{poly1} and @var{poly2}.
@example

(%i1) poly_add(z+x^2*y,x-z,[x,y,z]);
                                    2
(%o1)                              x  y + x
@end example

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_subtract (@var{poly1}, @var{poly2}, @var{varlist})
Subtracts a polynomial @var{poly2} from @var{poly1}.
@example

(%i1) poly_subtract(z+x^2*y,x-z,[x,y,z]);
                                      2
(%o1)                          2 z + x  y - x
@end example

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_multiply (@var{poly1}, @var{poly2}, @var{varlist})
Returns the product of polynomials @var{poly1} and @var{poly2}.
@example

(%i2) poly_multiply(z+x^2*y,x-z,[x,y,z])-(z+x^2*y)*(x-z),expand;
(%o1)                                  0
@end example

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_s_polynomial (@var{poly1}, @var{poly2}, @var{varlist})
Returns the @emph{syzygy polynomial} (@emph{S-polynomial}) of two polynomials @var{poly1} and @var{poly2}.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_primitive_part (@var{poly1}, @var{varlist})
Returns the polynomial @var{poly} divided by the GCD of its coefficients. 

@example
(%i1) poly_primitive_part(35*y+21*x,[x,y]);
(%o1)                              5 y + 3 x
@end example

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_normalize (@var{poly}, @var{varlist})
Returns the polynomial @var{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.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@subsection Other functions in grobner

@deffn {Function} poly_expand (@var{poly}, @var{varlist})
This function parses polynomials to internal form and back. It
is equivalent to @code{expand(@var{poly})} if @var{poly} parses correctly to
a polynomial. If the representation is not compatible with a
polynomial in variables @var{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.
@example

(%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

@end example

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_expt (@var{poly}, @var{number}, @var{varlist})
exponentitates @var{poly} by a positive integer @var{number}. If @var{number} is not a positive integer number an error will be raised.
@example

(%i1) poly_expt(x-y,3,[x,y])-(x-y)^3,expand;
(%o1)                                  0
@end example

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_content (@var{poly}. @var{varlist})
@code{poly_content} extracts the GCD of its coefficients
@example

(%i1) poly_content(35*y+21*x,[x,y]);
(%o1)                                  7
@end example

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_pseudo_divide (@var{poly}, @var{polylist}, @var{varlist})
Pseudo-divide a polynomial @var{poly} by the list of @math{n} polynomials @var{polylist}. Return
multiple values. The first value is a list of quotients @math{a}. The
second value is the remainder @math{r}. The third argument is a scalar
coefficient @math{c}, such that @math{c*poly} can be divided by @var{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:

@iftex
@tex
$$c*poly=\sum_{i=1}^{n}({a}_{i}*{polylist}_{i})+r$$
@end tex
@end iftex
@ifnottex
@math{c*poly=sum(a[i]*polylist[i],i=1...n)+r}.
@end ifnottex

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_exact_divide (@var{poly1}, @var{poly2}, @var{varlist})
Divide a polynomial @var{poly1} by another polynomial @var{poly2}. Assumes that exact
division with no remainder is possible. Returns the quotient.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_normal_form (@var{poly}, @var{polylist}, @var{varlist})
@code{poly_normal_form} finds the normal form of a polynomial @var{poly} with respect
to a set of polynomials @var{polylist}.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_buchberger_criterion (@var{polylist}, @var{varlist})
Returns @code{true} if @var{polylist} is a Groebner basis with respect to the current term
order, by using the Buchberger
criterion: for every two polynomials @math{h1} and @math{h2} in @var{polylist} the
S-polynomial @math{S(h1,h2)} reduces to 0 @math{modulo} @var{polylist}.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_buchberger (@var{polylist_fl} @var{varlist})
@code{poly_buchberger} performs the Buchberger algorithm on a list of
polynomials and returns the resulting Groebner basis.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn


@subsection Standard postprocessing of Groebner Bases

@iftex
@tex
The \emph{k-th elimination ideal} $I_k$ of an ideal $I$ over 
$K [ x_1, ...,x_1 ]$ is $I \cap K [ x_{k + 1}, ..., x_n ]$.

\noindent
The \emph{colon ideal} $I : J$ is the ideal $\{ h|\forall w \in J : wh \in
I \}$.@*

\noindent
The ideal $I : p^{\infty}$ is the ideal $\{ h|\exists n \in N : p^n h \in I \}$.@*

\noindent
The ideal $I : J^{\infty}$ is the ideal $\{ h|\exists n \in N, \exists p \in J: p^n h \in I \}$.@*

\noindent
The \emph{radical ideal} $\sqrt{I}$ is the ideal $\{ h| \exists n \in N :
h^n \in I \}$.@*

@end tex
@end iftex

@ifnottex
The @emph{k-th elimination Ideal} @math{I_k} of an Ideal @math{I} over @math{K[ x[1],...,x[n] ]} is the ideal @math{intersect(I, K[ x[k+1],...,x[n] ])}.@*
@noindent
The @emph{colon ideal} @math{I:J} is the ideal @math{@{h|for all w in J: w*h in I@}}.@*
@noindent
The ideal @math{I:p^inf} is the ideal @math{@{h| there is a n in N: p^n*h in I@}}.@*
@noindent
The ideal @math{I:J^inf} is the ideal @math{@{h| there is a n in N and a p in J: p^n*h in I@}}.@*
@noindent
The @emph{radical ideal} @math{sqrt(I)} is the ideal 
@math{@{h| there is a n in N : h^n in I @}}.
@end ifnottex

@noindent
@deffn {Function} poly_reduction (@var{polylist}, @var{varlist})
@code{poly_reduction} reduces a list of polynomials @var{polylist}, so that
each polynomial is fully reduced with respect to the other polynomials.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_minimization (@var{polylist}, @var{varlist})
Returns a sublist of the polynomial list @var{polylist} spanning the same
monomial ideal as @var{polylist} but minimal, i.e. no leading monomial
of a polynomial in the sublist divides the leading monomial
of another polynomial.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn


@deffn {Function} poly_normalize_list (@var{polylist}, @var{varlist})
@code{poly_normalize_list} applies @code{poly_normalize} to each polynomial in the list.
That means it divides every polynomial in a list @var{polylist} by its leading coefficient.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_grobner (@var{polylist}, @var{varlist})
Returns a Groebner basis of the ideal span by the polynomials @var{polylist}. Affected by the global flags.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_reduced_grobner (@var{polylist}, @var{varlist})
Returns a reduced Groebner basis of the ideal span by the polynomials @var{polylist}. Affected by the global flags.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn


@deffn {Function} poly_depends_p (@var{poly}, @var{var}, @var{varlist})
@code{poly_depends} tests whether a polynomial depends on a variable @var{var}.

@opencatbox
@category{Package grobner} @category{Predicate functions}
@closecatbox

@end deffn


@deffn {Function} poly_elimination_ideal (@var{polylist}, @var{number}, @var{varlist})


@code{poly_elimination_ideal} returns the grobner basis of the @math{number}-th elimination ideal of an
ideal specified as a list of generating polynomials (not necessarily Groebner basis).

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_colon_ideal (@var{polylist1}, @var{polylist2}, @var{varlist})

Returns the reduced Groebner basis of the colon ideal 

@math{I(polylist1):I(polylist2)}

@noindent
where @math{polylist1} and @math{polylist2} are two lists of polynomials.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_ideal_intersection (@var{polylist1}, @var{polylist2}, @var{varlist})

@code{poly_ideal_intersection} returns the intersection of two ideals.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn


@deffn {Function} poly_lcm (@var{poly1}, @var{poly2}, @var{varlist})
Returns the lowest common multiple of @var{poly1} and @var{poly2}.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@c -----------------------------------------------------------------------------
@anchor{poly_gcd}
@deffn {Function} poly_gcd (@var{poly1}, @var{poly2}, @var{varlist})

Returns the greatest common divisor of @var{poly1} and @var{poly2}.

See also @mrefcomma{ezgcd} @mrefcomma{gcd} @mrefcomma{gcdex} and
@mrefdot{gcdivide}

Example:

@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
@end example

@opencatbox
@category{Package grobner}
@closecatbox
@end deffn

@deffn {Function} poly_grobner_equal (@var{polylist1}, @var{polylist2}, @var{varlist})
@code{poly_grobner_equal} tests whether two Groebner Bases generate the same ideal.
Returns @code{true} if two lists of polynomials @var{polylist1} and @var{polylist2}, assumed to be Groebner Bases,
generate the same ideal, and @code{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 @code{false}.

@example
(%i1) poly_grobner_equal([y+x,x-y],[x,y],[x,y]);
(%o1)                         false
@end example

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_grobner_subsetp (@var{polylist1}, @var{polylist2}, @var{varlist})

@code{poly_grobner_subsetp} tests whether an ideal generated by @var{polylist1}
is contained in the ideal generated by @var{polylist2}. For this test to always succeed,
@var{polylist2} must be a Groebner basis.

@opencatbox
@category{Package grobner} @category{Predicate functions}
@closecatbox

@end deffn

@deffn {Function} poly_grobner_member (@var{poly}, @var{polylist}, @var{varlist})

Returns @code{true} if a polynomial @var{poly} belongs to the ideal generated by the
polynomial list @var{polylist}, which is assumed to be a Groebner basis. Returns @code{false} otherwise.

@code{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 @code{normal_form} being 0. 

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_ideal_saturation1 (@var{polylist}, @var{poly}, @var{varlist})
Returns the reduced Groebner basis of the saturation of the ideal
@iftex
@tex
$$I(polylist):poly^\infty$$
@end tex
@end iftex

@ifnottex
I(polylist):poly^inf
@end ifnottex

@noindent
Geometrically, over an algebraically closed field, this is the set
of polynomials in the ideal generated by @var{polylist} which do not identically
vanish on the variety of @var{poly}.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_ideal_saturation (@var{polylist1}, @var{polylist2}, @var{varlist})
Returns the reduced Groebner basis of the saturation of the ideal
@iftex
@tex
$$I(polylist1):I(polylist2)^\infty$$
@end tex
@end iftex

@ifnottex
I(polylist1):I(polylist2)^inf
@end ifnottex

@noindent
Geometrically, over an algebraically closed field, this is the set of polynomials in
the ideal generated by @var{polylist1} which do not identically vanish on the
variety of @var{polylist2}.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_ideal_polysaturation1 (@var{polylist1}, @var{polylist2}, @var{varlist})
@var{polylist2} ist a list of n polynomials @code{[poly1,...,polyn]}.
Returns the reduced Groebner basis of the ideal
@iftex
@tex
$$I(polylist):poly1^\infty:...:polyn^\infty$$
@end tex
@end iftex

@ifnottex
I(polylist):poly1^inf:...:polyn^inf
@end ifnottex

@noindent
obtained by a
sequence of successive saturations in the polynomials
of the polynomial list @var{polylist2} of the ideal generated by the
polynomial list @var{polylist1}.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_ideal_polysaturation (@var{polylist}, @var{polylistlist}, @var{varlist})
@var{polylistlist} is a list of n list of polynomials @code{[polylist1,...,polylistn]}.
Returns the reduced Groebner basis of the saturation of the ideal
@iftex
@tex
$$I(polylist):I(polylist_1)^\infty:...:I(polylist_n)^\infty$$
@end tex
@end iftex

@ifnottex
I(polylist):I(polylist_1)^inf:...:I(polylist_n)^inf
@end ifnottex

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_saturation_extension (@var{poly}, @var{polylist}, @var{varlist1}, @var{varlist2})

@code{poly_saturation_extension} implements the famous Rabinowitz trick.

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

@deffn {Function} poly_polysaturation_extension (@var{poly}, @var{polylist}, @var{varlist1}, @var{varlist2})

@opencatbox
@category{Package grobner}
@closecatbox

@end deffn

