(* ---------------------------------------------------------------------------- * $Id: SACPGCD.md,v 1.2 1992/02/12 17:35:01 pesch Exp $ * ---------------------------------------------------------------------------- * This file is part of MAS. * ---------------------------------------------------------------------------- * Copyright (c) 1989 - 1992 Universitaet Passau * ---------------------------------------------------------------------------- * $Log: SACPGCD.md,v $ * Revision 1.2 1992/02/12 17:35:01 pesch * Moved CONST definition to the right place * * Revision 1.1 1992/01/22 15:15:43 kredel * Initial revision * * ---------------------------------------------------------------------------- *) DEFINITION MODULE SACPGCD; (* SAC Polynomial GCD and RES System Definition Module. *) FROM MASSTOR IMPORT LIST;CONSTrcsid = "$Id: SACPGCD.md,v 1.2 1992/02/12 17:35:01 pesch Exp $";CONSTcopyright = "Copyright (c) 1989 - 1992 Universitaet Passau";PROCEDURE IPC(RL,A: LIST): LIST; (*Integral polynomial content. A is an integral polynomial in r variables. C is the content of A.*)PROCEDURE IPCPP(RL,A: LIST;VARC,AB: LIST); (*Integral polynomial content and primitive part. A is an integral polynomial in r variables. C is the content of A and AB is the primitive part of A.*)PROCEDURE IPGCDC(RL,A,B: LIST;VARC,AB,BB: LIST); (*Integral polynomial greatest common divisor and cofactors. A and B are integral polynomials in r variables, r ge 0. C=GCD(A,B). If C is non-zero then AB=A/C and BB=B/C. Otherwise AB=0 and BB=0.*)PROCEDURE IPIC(RL,A: LIST): LIST; (*Integral polynomial integer content. A is an integral polynomial in r variables. c is the integer content of A.*)PROCEDURE IPICPP(RL,A: LIST;VARAL,AB: LIST); (*Integral polynomial integer content and primitive part. A is an integral polynomial in r variables. a is the integer content of A. AB=A/a if A is non-zero and AB=0 if A=0.*)PROCEDURE IPICS(RL,A,CL: LIST): LIST; (*Integral polynomial integer content subroutine. A is a non-zero integral polynomial in r variables. c is an integer. d is the greatest common divisor of c and the integer content of A.*)PROCEDURE IPIPP(RL,A: LIST): LIST; (*Integral polynomial integer primitive part. A is an integral polynomial in r variables. If A ne 0 then AB=A/a where a is the integer content of A. If A=0 then AB=0.*)PROCEDURE IPPGSD(RL,A: LIST): LIST; (*Integral polynomial primitive greatest squarefree divisor. A is an integral polynomial in r variables. If A=0 then B=0. Otherwise B is the greatest squarefree divisor of the primitive part of A.*)PROCEDURE IPPP(RL,A: LIST): LIST; (*Integral polynomial primitive part. A is an integral polynomial in r variables. AB is the primitive part of A.*)PROCEDURE IPRES(RL,A,B: LIST): LIST; (*Integral polynomial resultant. A and B are integral polynomials in r variables, r ge 1, of positive degrees. C is the resultant of A and B with respect to the main variable, an integral polynomial in r-1 variables.*)PROCEDURE IPRPRS(RL,A,B: LIST): LIST; (*Integral polynomial reduced polynomial remainder sequence. A and B are non-zero integral polynomials in r variables with deg(A) ge deg(B). S is the reduced polynomial remainder sequence of A and B.*)PROCEDURE IPSCPP(RL,A: LIST;VARSL,C,AB: LIST); (*Integral polynomial sign, content, and primitive part. A is an integral polynomial in R ge 1 variables. s is the sign of A, C is the content of A, and AB is the primitive part of A.*)PROCEDURE IPSF(RL,A: LIST): LIST; (*Integral polynomial squarefree factorization. A is a positive pri- mitive integral polynomial in r variables of positive degree. L is the list ((e sub 1,A sub 1), ...,(e sub l,A sub l)) where A equal to the product of (A sub i)**(e sub i) for i = 1, ...,k is the squarefree factorization of A in which 1 le e sub 1 lt e sub 2 lt ... lt e sub k and each A sub i is a positive squarefree polynomial of positive degree.*)PROCEDURE IPSPRS(RL,A,B: LIST): LIST; (*Integral polynomial subresultant polynomial remainder sequence. A and B are non-zero integral polynomials in r variables with deg(A) ge deg(B). S is the subresultant p.r.s. of the first kind of A and B.*)PROCEDURE IPSRP(RL,A: LIST;VARAL,AB: LIST); (*Integral polynomial similiar to rational polynomial. A is a rational polynomial in r variables, r ge 0. a is a rational number, and AB is an integral polynomial such that A=a*AB. If A eq 0 then a=AB=0. Otherwise AB is integer primitive and positive.*)PROCEDURE MPGCDC(RL,PL,A,B: LIST;VARC,AB,BB: LIST); (*Modular polynomial greatest common divisor and cofactors. p is a prime beta-integer. A and B are polynomials in r variables over Z sub p. C=gcd(A,B). If C is non-zero then AB=A/C and BB=B/C. Otherwise AB=0 and BB=0.*)PROCEDURE MPRES(RL,PL,A,B: LIST): LIST; (*Modular polynomial resultant. p is a prime beta-digit. A and B are polynomials over Z sub p in r variables, r ge 1, of positive degree. C is the resultant of A and B, a polynomial over Z sub p in r-1 variables.*)PROCEDURE MPSPRS(RL,PL,A,B: LIST): LIST; (*Modular polynomial subresultant polynomial remainder sequence. A and B are non-zero polynomials in r variables over Z sub p, p a prime beta-integer, with deg(A) ge deg(B). S is the subresultant p.r.s. of the first kind of A and B.*)PROCEDURE MPUC(RL,PL,A: LIST): LIST; (*Modular polynomial univariate content. A is a polynomial in r variables, r ge 2, over Z sub p, p a prime beta-integer. c is the univariate content of A.*)PROCEDURE MPUCPP(RL,PL,A: LIST;VARAL,AB: LIST); (*Modular polynomial univariate content and primitive part. A is a polynomial in r variables, r ge 2, over Z sub p, p a prime beta-integer. a is the univariate content of A. AB=A/a if A is non-zero and AB=0 if A=0.*)PROCEDURE MPUCS(RL,PL,A,CL: LIST): LIST; (*Modular polynomial univariate content subroutine. A is a non-zero polynomial in r variables, r ge 2, over Z sub p, p a prime beta-integer. c is a univariate polynomial over Z sub p. d is the greatest common divisor of c and the univariate content of A.*)PROCEDURE MPUPP(RL,PL,A: LIST): LIST; (*Modular polynomial univariate primitive part. A is a polynomial in r variables, r ge 2, over Z sub p, p a prime beta-integer. If A is non-zero then AB=A/a where a is the univariate content of A. If A=0 then AB=0.*)PROCEDURE MUPEGC(PL,A,B: LIST;VARC,U,V: LIST); (*Modular univariate polynomial extended greatest common divisor. p is a prime beta-integer. A and B are univariate polynomials over Z sub p. C=gcd(A,B). A*U+B*V=C, and, if deg(A/C) gt 0, then deg(V) lt deg(A/C), else deg(V)=0. Similarly, if deg(B/C) gt 0, then deg(U) lt deg(B/C), else deg(U)=0. If A=0, U=0. If B=0, V=0.*)PROCEDURE MUPGCD(PL,A,B: LIST): LIST; (*Modular univariate polynomial greatest common divisor. A and B are univariate polynomials over Z sub p, p a prime beta-integer. C=gcd(A,B).*)PROCEDURE MUPHEG(PL,A,B: LIST;VARC,V: LIST); (*Modular univariate polynomial half-extended greatest common divisor. p is a prime beta-integer. A and B are univariate polynomials over Z sub p. C=gcd(A,B). There exists a polynomial U such that A*U+B*V=C, and, if deg(A/C) gt 0, then deg(V) lt deg(A/C). If deg(A/C)=0, deg(V) is also 0. If B=0, V=0.*)PROCEDURE MUPRES(PL,A,B: LIST): LIST; (*Modular univariate polynomial resultant. p is a prime beta-digit. A and B are univariate polynomials over Z sub p of positive degrees. C is the resultant of A and B, an element of Z sub p.*)PROCEDURE MUPSFF(PL,A: LIST): LIST; (*Modular univariate polynomial squarefree factorization. p is a prime beta-integer. A is a monic univariate polynomial over Z sub p of positive degree. L is a list ((i(1),A(1)), ...,(i(r),A(r))) with i(1) lt i(2) lt ... lt i(r), A(j) a monic squarefree factor of a of positive degree for 1 le j le r and A the product of A(j)**i(j) for j=1, ...,r.*)PROCEDURE RPBLGS(RL,A: LIST;VARAL,BL,SL: LIST); (*Rational polynomial base coefficients least common multiple, greatest common divisor, and sign. A is a rational polynomial in r variables, r ge 0. If A=0 then a=b=s=0. Otherwise, a is the lcm of the denominators of the base coefficients of A, b is the gcd of the numerators of the base coefficients of A, and s is the sign of the leading base coefficient of A.*)ENDSACPGCD. (* -EOF- *)