1/* -*- Mode: Maxima -*- */ 2 3/* 4** 5** $Id: grobner.demo,v 1.2 2003-05-03 11:40:00 starseeker Exp $ 6** Copyright (C) 1999, 2002 Marek Rychlik <rychlik@u.arizona.edu> 7** 8** This program is free software; you can redistribute it and/or modify 9** it under the terms of the GNU General Public License as published by 10** the Free Software Foundation; either version 2 of the License, or 11** (at your option) any later version. 12** 13** This program is distributed in the hope that it will be useful, 14** but WITHOUT ANY WARRANTY; without even the implied warranty of 15** MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 16** GNU General Public License for more details. 17** 18** You should have received a copy of the GNU General Public License 19** along with this program; if not, write to the Free Software 20** Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. 21** 22*/ 23showtime:true; 24 25/* POLY_MONOMIAL_ORDER switch represents the monomial order that will globally be in effect 26 for the succeeding operations. */ 27 28poly_monomial_order:'lex; 29 30/* POLY_EXPAND parses polynomials to internal form and back. It can be used to test 31 whether an expression correctly parses to the internal representation. 32 The following examples illustrate that indexed and transcendental function variables 33 are allowed. */ 34 35poly_expand(x,[x,y]); 36poly_expand(x+y,[x,y]); 37poly_expand(x-y,[x,y]); 38poly_expand((x-y)*(x+y),[x,y]); 39poly_expand((x+y)^2,[x,y]); 40poly_expand((x+y)^5,[x,y]); 41poly_expand(x/y-1,[x]); 42poly_expand(x^2/sqrt(y)-x*exp(y)-1,[x]); 43poly_expand(sin(x)-sin(x)^2-1,[sin(x)]); 44poly_expand((x[2]/sin(y[3])-1)^5,[x[2]]),poly_return_term_list:true; 45 46/* POLY_ADD, POLY_SUBTRACT, POLY_MULTIPLY and POLY_EXPT are the arithmetical operations on polynomials. 47 These are performed using the internal representation, but the results are converted back to the 48 Maxima general form */ 49 50poly_add(x^2*y+z,x-z,[x,y,z]); 51poly_subtract(x^2*y+z,x-z,[x,y,z]); 52poly_multiply(x^2*y+z,x-z,[x,y,z]) - (x^2*y+z)*(x-z), expand; 53poly_expt(x-y, 3, [x,y]) - (x-y)^3, expand; 54 55/* POLY_CONTENT extracts the GCD of its coefficients */ 56poly_content(21*x+35*y,[x,y]); 57 58/* POLY_PRIMITIVE_PART divides a polynomial by the GCD of its coefficients */ 59poly_primitive_part(21*x+35*y,[x,y]); 60 61/* POLY_S_POLYNOMIAL computest the syzygy polynomial (S-polynomial) of two polynomials */ 62poly_s_polynomial(x+y,x-y,[x,y]); 63 64 65/* POLY_NORMAL_FORM finds the normal form of a polynomial with respect to a set of polynomials. */ 66poly_normal_form(x^2+y^2,[x-y,x+y],[x,y]); 67poly_pseudo_divide(2*x^2+3*y^2,[7*x-y^2,11*x+y],[x,y]); 68poly_exact_divide((x+y)^2,x+y,[x,y]); 69 70/* POLY_BUCHBERGER performs the Buchberger algorithm on a list of polynomials and returns 71 the resulting Grobner basis */ 72poly_buchberger([x^2-y*x,x^2+y+x*y^2],[x,y]); 73 74/* POLY_REDUCTION reduces a set of polynomials, so that 75 each polynomial is fully reduced with respect to the other polynomials */ 76 77poly_reduction([x^2-x*y,x*y^2+y+x^2,x*y^2+x*y+y,x*y-y^2,y^3+y^2+y],[x,y]); 78 79/* POLY_MINIMIZATION selects a subset of a set of polynomials, so that no leading monomial is divisible by 80 another leading monomial */ 81 82poly_minimization([x^2-x*y,x*y^2+y+x^2,x*y^2+x*y+y,x*y-y^2,y^3+y^2+y],[x,y]); 83 84/* POLY_REDUCED_GROBNER returns a reduced Grobner basis */ 85poly_reduced_grobner([x^2-y*x,x^2+y+x*y^2],[x,y]); 86 87/* POLY_NORMALIZE divides a polynomial by its leading coefficient */ 88poly_normalize(2*x+y,[x,y]); 89 90/* POLY_NORMALIZE_LIST applies POLY_NORMALIZE to each polynomial in the list */ 91 92poly_normalize_list([2*x+y,3*x^2+7],[x,y]); 93 94/* POLY_DEPENDS_P tests whether a polynomial depends on a variable */ 95 96poly_depends_p(x^2+y,x,[x,y,z]); 97poly_depends_p(x^2+y,z,[x,y,z]); 98 99 100/* POLY_ELIMINATION_IDEAL returns the grobner basis of the K-th elimination ideal of an 101 ideal specified as a list of generating polynomials (not necessarily Grobner basis */ 102 103poly_elimination_ideal([x+y,x-y],0,[x,y]); 104poly_elimination_ideal([x+y,x-y],1,[x,y]); 105poly_elimination_ideal([x+y,x-y],2,[x,y]); 106 107/* POLY_IDEAL_INTERSECTION returns the intersection of two ideals */ 108poly_ideal_intersection([x^2+y,x^2-y],[x,y^2],[x,y]); 109 110/* POLY_LCM and POLY_GCD are the Grobner versions of LCM and GCD */ 111 112poly_lcm(x*y^2-x,x^2*y+x,[x,y]); 113poly_gcd(x*y^2-x,x^2*y+x,[x,y]); 114 115/* POLY_GROBNER_MEMBER tests whether a polynomial belongs to an ideal generated by a list of polynomials, 116 which is assumed to be a Grobner basis. Equivalent to NORMAL_FORM being 0. */ 117 118poly_grobner_member(x+y,[x,y],[x,y]); 119 120/* POLY_GROBNER_EQUAL tests whether two Grobner bases generate the same ideal. 121 This is equivalent to checking that every polynomial of the first basis reduces to 0 122 modulo the second basis and vice versa. Note that in the example below the 123 first list is not a Grobner basis, and thus the result is FALSE. */ 124 125poly_grobner_equal([x+y,x-y],[x,y],[x,y]); 126 127/* POLY_GROBNER_SUBSETP tests whether an ideal generated by the first list of polynomials 128 is contained in the ideal generated by the second list. For this test to always succeed, 129 the second list must be a Grobner basis */ 130 131poly_grobner_subsetp([x+y,x-y],[x,y],[x,y]); 132 133/* POLY_POLYSATURATION_EXTENSION implements the famous Rabinowitz trick. */ 134poly_polysaturation_extension([x,y],[x^2,y^3],[x,y],[u,v]); 135 136poly_saturation_extension([x,y],[x^2,y^3],[x,y],[u,v]); 137 138/* POLY_IDEAL_POLYSATURATION1 for a given ideal I and polynomials f, g, ..., finds 139 the colon ideal I : f^inf : g^inf : ... */ 140poly_ideal_polysaturation1([x,y],[x^2,y^3],[x,y]); 141 142/* POLY_IDEAL_SATURATION for given ideals I and J computes the ideal I : J^inf. */ 143poly_ideal_saturation([x,y],[x^2,y^3],[x,y]); 144 145/* POLY_IDEAL_POLYSATURATION for a given ideal I and a sequence of ideals J1, J2, J3, ..., 146 finds the ideal I : J1^inf : J2^inf : J3^inf : ... */ 147poly_ideal_polysaturation([x,y],[[x^2],[y^3]],[x,y]); 148poly_ideal_polysaturation([x^4-y^4], [[x-y],[x^2+y^2, x+y]],[x,y]); 149 150/* POLY_COLON_IDEAL finds the reduced Grobner basis of the colon ideal I:J, i.e. the set of polynomials h 151 such that there is a polynomial F in J for which H*F is in I */ 152 153poly_colon_ideal([x^2*y],[y],[x,y]); 154 155/* POLY_BUCHBERGER_CRITERION verifies whether a given set of polynomials is a Grobner basis with respect 156 to the current term order */ 157poly_buchberger_criterion([x,y],[x,y]); 158poly_buchberger_criterion([x-y,x+y],[x,y]); 159 160/* Grobner basis associated with Enneper minimal surface */ 161poly_grobner([x-3*u-3*u*v^2+u^3,y-3*v-3*u^2*v+v^3,z-3*u^2+3*v^2],[u,v,x,y,z]); 162poly_reduced_grobner([x-3*u-3*u*v^2+u^3,y-3*v-3*u^2*v+v^3,z-3*u^2+3*v^2],[u,v,x,y,z]); 163 164/* Cyclic roots of degree 5 */ 165poly_reduced_grobner([x+y+z+u+v,x*y+y*z+z*u+u*v+v*x,x*y*z+y*z*u+z*u*v+u*v*x+v*x*y,x*y*z*u+y*z*u*v+z*u*v*x+u*v*x*y+v*x*y*z,x*y*z*u*v-1],[u,v,x,y,z]); 166 167/* The next example demonstrates the use of the switch 168 POLY_RETURN_TERM_LIST, which, if set to TRUE, makes the results to 169 appear as lists of terms listed in the current monomial order rather 170 than a general form expression */ 171 172block([orders:[lex,grlex,grevlex,invlex]], 173for i:1 thru length(orders) do ( 174 print(ev([orders[i], poly_expand((x^2+x+y)^3,[x,y])], poly_monomial_order=orders[i])) 175 ) 176), poly_return_term_list=true; 177 178/* Grobner bases can be computed over coefficient ring of maxima general expressions */ 179poly_grobner([x*y-1,x+y],[x]); 180 181/* A tough example learned from Cox */ 182poly_grobner([x^5+y^4+z^3-1,x^3+y^3+z^2-1], [x,y,z]); 183 184/* An even tougher example of Cox */ 185poly_grobner([x^5+y^4+z^3-1,x^3+y^3+z^2-1], [x,y,z]); 186 187/* We can also perform Grobner basis calculations modulo prime */ 188poly_grobner([x^5+y^4+z^3-1,x^3+y^3+z^2-1], [x,y,z]), modulus=3; 189 190/* We can also explicitly ask for the Grobner basis to be calculated using only 191 integer coefficients. An error will result if this assertion is not satisfied. */ 192poly_grobner([x^5+y^4+z^3-1,x^3+y^3+z^2-1], [x,y,z]), poly_coefficient_ring='ring_of_integers; 193 194/* The following several tests demonstrate the use of jet variables useful in processing differential equations */ 195 196 197/* Clear some variables */ 198kill(ode,t,x,y,u,v,r); 199 200/* Set up dependencies */ 201depends([x,y,u,v,r],t); 202 203/* These are equations representing mathematical pendulum */ 204ode:[x^2+y^2-c,'diff(x,t)-u,'diff(y,t)-v,'diff(u,t)+r*x,'diff(v,t)+r*y+1]; 205 206jet_vars(k):=apply(append,reverse(makelist(['diff(x,t,j),'diff(y,t,j),'diff(u,t,j),'diff(v,t,j),'diff(r,t,j)],j,0,k+1))); 207 208/* Define k-fold prolongation */ 209prolongate(k):=apply(append,makelist(diff(ode,t,j),j,0,k)); 210 211/* Define Grobner basis of k-fold prolongation */ 212gb(k):=poly_reduced_grobner(prolongate(k),jet_vars(k)); 213 214/* Define the l-th projection of the k-th prolongation */ 215projection(l, k):=poly_elimination_ideal(prolongate(k),5*l,jet_vars(k)); 216 217/* Compute some projections */ 218projection(0, 0); 219projection(1, 1); 220