This is the cl-buchberger Reference Manual, version 0.0.4,
generated automatically by Declt version 4.0 beta 2 "William Riker"
on Wed Jun 15 03:32:35 2022 GMT+0.
1 Introduction
cl-buchberger
=============
Juan M. Bello Rivas
June 2009
image::images/cl-buchberger.png[Screenshot]
Overview
--------
cl-buchberger is a Common Lisp implementation of Buchberger's
algorithm for the computation of Gröbner bases.
Currently this package can compute Gröbner bases of ideals in
multivariate polynomial rings over the rationals.
Availability
------------
Download the latest version of this package from
http://github.com/jmbr/cl-buchberger/tarball/master
Portability
-----------
The code has been tested under
1. ABCL 0.16.0-dev
2. CLISP 2.47
3. ECL 9.6.1
4. SBCL 1.0.29
License
-------
cl-buchberger is released under the GNU GPLv2.
Usage
-----
Loading the package
~~~~~~~~~~~~~~~~~~~
You need ASDF installed in your system. To load cl-buchberger,
write:
------------------------------------------------------
CL-USER> (asdf:operate 'asdf:load-op :cl-buchberger)
NIL
CL-USER> (use-package :cl-buchberger)
T
------------------------------------------------------
Defining a polynomial ring
~~~~~~~~~~~~~~~~~~~~~~~~~~
There's a default ring which is the ring of polynomials on X, Y, Z
over the rationals. To define custom polynomial rings use:
---------------------------------------------------------
CL-USER> (make-instance 'polynomial-ring :variables
(list 'x 'y 'z 'u 'w 'r 's 't))
#
---------------------------------------------------------
To change the default ring just bind \*RING\* to whatever you want:
---------------------------------------------------------------------
CL-USER> (defparameter *ring*
(make-instance 'polynomial-ring :variables (list 'x 'y))
"QQ[X, Y]")
*RING*
CL-USER> *ring*
#
---------------------------------------------------------------------
Defining polynomials
~~~~~~~~~~~~~~~~~~~~
Polynomials are defined using a list of terms where each term is a
list with the coefficient as its first term and where the remaining
elements are variable exponents. For example: (1 1 2 3) would
correspond to the term $$`xy^2z^3`$$
Thus, to create a polynomial write:
-------------------------------------------------------------------
CL-USER> (defparameter *l* (make-polynomial '((1 3 0) (-2 1 1))))
*L*
CL-USER> *l*
#
CL-USER> (defparameter *m* (make-polynomial '((3 4 1) (1 2 0))))
*M*
CL-USER> *m*
#
-------------------------------------------------------------------
Polynomial arithmetic
~~~~~~~~~~~~~~~~~~~~~
Use the generic functions RING+, RING-, RING*, and RING/ for the usual
arithmetic operations.
The function RING/ is the multivariate polynomial division algorithm
and takes a polynomial and a sequence of divisors to produce a
sequence of quotients and a remainder.
To set the default monomial ordering, bind \*MONOMIAL-ORDERING\* to the
relevant function (which defaults to LEX>). For example:
--------------------------------------------------------
CL-USER> (defparameter *monomial-ordering* #'grevlex>)
*MONOMIAL-ORDERING*
--------------------------------------------------------
Also, you can use the macro WITH-MONOMIAL-ORDERING to define the
current monomial ordering as in:
-------------------------------------------
CL-USER> (with-monomial-ordering #'grlex>
(ring/ *m* *l*))
#(#)
#
-------------------------------------------
Computing Gröbner bases
~~~~~~~~~~~~~~~~~~~~~~~
The functions GROEBNER and REDUCED-GROEBNER compute Gröbner bases and
reduced Gröbner bases respectively. Both of them take a sequence of
polynomials as parameter. An alternative is to construct a polynomial
ideal and obtain its reduced Gröbner basis using the BASIS generic
function.
For example:
--------------------------------------------------------------------------------
CL-USER> (defparameter *katsura-3*
(make-ideal (list (make-polynomial '((1 1 0 0) (2 0 1 0) (2 0 0 1) (-1 0 0 0)))
(make-polynomial '((1 2 0 0) (-1 1 0 0) (2 0 2 0) (2 0 0 2)))
(make-polynomial '((2 1 1 0) (2 0 1 1) (-1 0 1 0)))))
"Katsura-3 over QQ[x, y, z] (default ring)")
*KATSURA-3*
CL-USER> *katsura-3*
# :GENERATORS #(#
#
#)>
CL-USER> (basis *katsura-3*)
#(#
#
#)
--------------------------------------------------------------------------------
Bug reports
-----------
There's a bug tracker available at http://github.com/jmbr/cl-buchberger/issues