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AFUPGC.c
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AFUPGC.c
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/*===========================================================================
AFUPGC(M,A,B; C,Ab,Bb)
Algebraic number field univariate polynomial greatest common divisor
and cofactors.
Inputs
M : in Z[x], the minimal polynomial of an algebraic number alpha.
A,B : in Q(alpha)[x].
Outputs
C : in Q(alpha)[x], C is the monic gcd of A and B.
Ab,Bb : in Q(alpha)[x], the cofactors Ab = A / C and Bb = B / C.
===========================================================================*/
#include "saclib.h"
void AFUPGC(M,A,B,C_,Ab_,Bb_)
Word M,A,B,*C_,*Ab_,*Bb_;
{
Word A1,A1p,A2,A2p,Ab,Bb,C,Cp,Ct,D,G,L,Lt,Mp,Mpp,P,Q,Qp,Qt,Qtp,R,T,
c,d,p,q,qp,qt,t;
Word *Dp,*Sp,*Spp;
Step1: /* A = B = 0. */
if (A == 0 && B == 0) {
C = 0;
Ab = 0;
Bb = 0;
goto Return; }
Step2: /* A = 0 or B = 0. */
if (A == 0) {
C = AFPMON(1,M,B);
Ab = 0;
Bb = PINV(0,PLDCF(B),1);
goto Return; }
if (B == 0) {
C = AFPMON(1,M,A);
Bb = 0;
Ab = PINV(0,PLDCF(A),1);
goto Return; }
Step3: /* deg A = 1 or deg B = 1. */
if (PDEG(A) == 1 || PDEG(B) == 1) {
AFUPGC1(M,A,B,&C,&Ab,&Bb);
goto Return; }
/* General case. */
Step4: /* Initialize. */
P = MPRIME;
A1 = AFPICR(1,A);
A2 = AFPICR(1,B);
IPSRP(2,A1,&R,&A1);
IPSRP(2,A2,&R,&A2);
Q = 1;
C = 0;
Ct = 0;
Qt = 1;
q = 1;
L = NIL;
d = PDEG(B) + 1;
Step5: /* Reduce polynomials modulo a prime. */
if (P == NIL)
FAIL("AFUPGCD","Prime list exhausted");
ADV(P,&p,&P);
Mp = MPHOM(1,p,M);
if (PDEG(Mp) < PDEG(M))
goto Step5;
A1p = MPHOM(2,p,A1);
if (PDEG(A1p) < PDEG(A1))
goto Step5;
A2p = MPHOM(2,p,A2);
if (PDEG(A2p) < PDEG(A2))
goto Step5;
Step6: /* Check if p divides disc(M). */
Mpp = MUPDER(p,Mp);
Sp = MAPFMUP(Mp);
Spp = MAPFMUP(Mpp);
Dp = MMAPGCD(p,Sp,Spp);
D = MUPFMAP(Dp);
MAPFREE(Sp);
MAPFREE(Spp);
MAPFREE(Dp);
if (PDEG(D) > 0)
goto Step5;
Step7: /* Compute monic gcd modulo p. */
Mp = MPMON(1,p,Mp);
Cp = FRUPGCD(p,Mp,A1p,A2p);
if (Cp == NIL)
goto Step5;
Step8: /* Check degree. */
c = PDEG(Cp);
if (c == 0) /* gcd is 1. */
goto Step12;
if (c > d) /* degree too large, throw prime away */
goto Step5;
if (c < d) { /* all previous primes were unlucky */
C = Cp;
d = c;
Q = p;
Qp = IQ(Q,2);
ISQRT(Qp,&Qp,&t);
q = ILOG2(Qp);
L = NIL;
goto Step5; }
Step9: /* Apply Chinese remaindering. */
if (Q != p) {
qp = MDINV(p,MDHOM(p,Q));
Qt = IDPR(Q,p);
Ct = IPCRA(Q,p,qp,2,C,Cp); }
Step10: /* Recover rational coefficients, if possible. */
Qtp = IQ(Qt,2);
ISQRT(Qtp,&Qtp,&t);
qt = ILOG2(Qtp);
AFUPFMRC(Q,Qp,q,C,L,Qt,Qtp,qt,Ct, &Lt,&G);
Q = Qt;
Qp = Qtp;
q = qt;
C = Ct;
L = Lt;
if (G == NIL)
goto Step5;
Step11: /* Trial division. */
AFUPQR(M,A,G,&Ab,&T);
if (T != 0)
goto Step5;
else {
AFUPQR(M,B,G,&Bb,&T);
if (T != 0)
goto Step5; }
C = G;
goto Return;
Step12: /* Relatively prime. */
C = PMON(AFFINT(1),0);
Ab = A;
Bb = B;
Return: /* Prepare for return. */
*C_ = C;
*Ab_ = Ab;
*Bb_ = Bb;
}