File: [local] / OpenXM / src / asir-contrib / testing / noro / ndbf.rr (download)
Revision 1.20, Fri Sep 5 11:55:19 2014 UTC (9 years, 9 months ago) by ohara
Branch: MAIN
CVS Tags: HEAD
Changes since 1.19: +7 -7
lines
Fixed for calling qsort, mapat because of recent changes in module implementation.
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/* $OpenXM: OpenXM/src/asir-contrib/testing/noro/ndbf.rr,v 1.20 2014/09/05 11:55:19 ohara Exp $ */
/* requires 'primdec' */
#define TMP_H hhhhhhhh
#define TMP_S ssssssss
#define TMP_DS dssssssss
#define TMP_T tttttttt
#define TMP_DT dtttttttt
#define TMP_Y1 yyyyyyyy1
#define TMP_DY1 dyyyyyyyy1
#define TMP_Y2 yyyyyyyy2
#define TMP_DY2 dyyyyyyyy2
if (!module_definedp("gr")) load("gr")$ else{ }$
if (!module_definedp("primdec")) load("primdec")$ else{ }$
if (!module_definedp("newsyz")) load("noro_module_syz.rr")$ else{ }$
/* Empty for now. It will be used in a future. */
/* toplevel */
module ndbf$
/* bfunction */
localf bfunction, in_ww, in_ww_main, ann, ann_n$
localf ann0, ann_fa, psi, ww_weight, compare_first, generic_bfct$
localf generic_bfct_1, initial_part, bfct, indicial1, bfct_via_gbfct$
localf bfct_via_gbfct_weight, bfct_via_gbfct_weight_1, bfct_via_gbfct_weight_2$
localf weyl_minipolym, weyl_minipoly, weyl_nf, weyl_nf_quo_check$
localf weyl_nf_quo, weyl_nf_mod, b_subst, v_factorial, w_tdeg$
localf replace_vars_f, replace_vars_v, replace_var$
localf action_on_gfs, action_on_gfs_1$
localf nd_gb_candidate$
localf in_gb_oaku, homogenize_oaku$
/* stratification */
localf weyl_subst, bfactor, gen_a, gen_d$
localf gen_dm, elimination, weyl_ideal_quotient, psi0$
localf bf_strat, bf_strat_stage2, bf_strat_stage3, bf_local$
localf conv_tdt, merge_tower, stratify_bf, tdt_homogenize$
localf sing, tower_in_p, subst_vars, compute_exponent$
localf zero_inclusion, weyl_divide_by_right, elim_mat, int_cons$
localf ideal_intersection$
def bfunction(F)
{
if ( member(s,vars(F)) )
error("ann : the variable 's' is reserved.");
/* F -> F/Fcont */
F1 = ptozp(F); Fcont = sdiv(F,F1); F = F1;
if ( type(Heu=getopt(heuristic)) == -1 ) Heu = 0;
if ( type(Vord=getopt(vord)) == -1 || type(Vord) != 4 ) Vord = 0;
if ( type(Wt=getopt(weight)) == -1 ) Wt = 0;
if ( type(Op=getopt(op)) == -1 ) Op = 0;
L = in_ww(F|weight=Wt,heuristic=Heu,vord=Vord);
Indata = L[0]; AllData = L[1]; VData = L[2];
GIN = Indata[0]; VDV = Indata[1]; WVDV = AllData[4];
W = Indata[4];
dp_set_weight(W);
B = weyl_minipoly(GIN,VDV,0,WVDV);
dp_set_weight(0);
if ( !Op ) return subst(B,s,-s-1);
V0 = VData[0]; DV0 = VData[1]; T = VData[2]; DT = VData[3];
BPT = weyl_subst(B,T*DT,VDV);
/* computation using G0,GIN0,VDV0 */
G0 = AllData[0]; GIN0 = AllData[1]; VDV0 = AllData[2]; WtV0 = AllData[5];
dp_set_weight(WtV0); dp_ord(0);
PS = map(dp_ptod,GIN0,VDV0); Len = length(PS);
for ( I = Len-1, Ind = []; I >= 0; I-- ) Ind = cons(I,Ind);
/* QR = [D,M,Coef] */
Ax = 1;
AxBPT = dp_ptod(Ax*BPT,VDV0);
QR = weyl_nf_quo(Ind,AxBPT,1,PS);
if ( !weyl_nf_quo_check(AxBPT,PS,QR) )
error("bfunction : invalid quotient");
if ( QR[0] ) error("bfunction : invalid quotient");
Den = QR[1]; Coef = QR[2];
for ( I = 0, R = Den*AxBPT; I < Len; I++ )
R -= dp_weyl_mul(Coef[I],dp_ptod(G0[I],VDV0));
R = dp_dtop(R,VDV0);
CR = conv_tdt(R,F,V0,DV0,T,DT);
dp_set_weight(0);
Cont = cont(CR); CR /= Cont;
Cont *= dn(Fcont); Den *= nm(Fcont);
Gcd = igcd(Den,Cont);
return [subst(B,s,-s-1),(Cont*CR)/(Den*Ax)];
}
/*
returns [InData,AllData,VData]
InData = [GIN,VDV,V,DV,WtV]
AllData = [G0,GIN0,VDV0,W,WVDV,WtV0]
VData = [V0,DV0,T,DT]
GIN0 = ini_(-W,W)(G0)
WVDV = W[0]*V[0]*DV[0]+...
*/
def in_ww(F)
{
F = ptozp(F);
V = vars(F);
N = length(V);
D = newvect(N);
Wt = getopt(weight);
Vord = getopt(vord);
if ( type(Wt) != 4 ) {
if ( type(Vord) != 4 ) {
for ( I = 0; I < N; I++ )
D[I] = [deg(F,V[I]),V[I]];
qsort(D,ndbf.compare_first);
for ( V = [], I = 0; I < N; I++ )
V = cons(D[I][1],V);
V = reverse(V);
} else
V = Vord;
for ( I = 0, Wt = []; I < N; I++ )
Wt = cons(1,Wt);
} else {
Wt1 = vector(N);
if ( type(Vord) != 4 ) {
for ( I = 0, F1 =F; I < N; I++ ) {
VI = Wt[2*I]; WI = Wt[2*I+1];
for ( J = 0; J < N; J++ )
if ( VI == V[J] ) break;
F1 = subst(F1,VI,VI^WI);
}
for ( I = 0; I < N; I++ )
D[I] = [deg(F1,V[I]),V[I]];
qsort(D,ndbf.compare_first);
for ( V = [], I = 0; I < N; I++ )
V = cons(D[I][1],V);
V = reverse(V);
} else
V = Vord;
for ( I = 0; I < N; I++ ) {
VI = Wt[2*I]; WI = Wt[2*I+1];
for ( J = 0; J < N; J++ )
if ( VI == V[J] ) break;
Wt1[J] = WI;
}
Wt = vtol(Wt1);
}
Tdeg = w_tdeg(F,V,Wt);
/* weight for [t,x1,...,xn,dt,dx1,...,dxn,h] */
WtV = newvect(2*(N+1)+1);
WtV[0] = Tdeg; WtV[N+1] = 1; WtV[2*(N+1)] = 1;
/* wdeg(V[I])=Wt[I], wdeg(DV[I])=Tdeg-Wt[I]+1 */
for ( I = 1; I <= N; I++ ) {
WtV[I] = Wt[I-1]; WtV[N+1+I] = Tdeg-Wt[I-1]+1;
}
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
B = [TMP_T-F];
for ( I = 0; I < N; I++ ) {
B = cons(DV[I]+diff(F,V[I])*TMP_DT,B);
}
V1 = cons(TMP_T,V); DV1 = cons(TMP_DT,DV);
W = newvect(N+1); W[0] = 1;
VW1 = [V1,DV1,WtV,W];
/* weight for [x1,...,xn,t,dx1,...,dxn,dt,h] */
WtV = newvect(2*(N+1)+1);
WtV[N] = Tdeg; WtV[2*N+1] = 1; WtV[2*(N+1)] = 1;
for ( I = 0; I < N; I++ ) {
WtV[I] = Wt[I]; WtV[N+1+I] = Tdeg-Wt[I]+1;
}
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
B = [TMP_T-F];
for ( I = 0; I < N; I++ ) {
B = cons(DV[I]+diff(F,V[I])*TMP_DT,B);
}
V1 = append(V,[TMP_T]); DV1 = append(DV,[TMP_DT]);
W = newvect(N+1); W[N] = 1;
VW2 = [V1,DV1,WtV,W];
Heu = getopt(heuristic);
if ( type(Heu) != -1 && Heu )
L = in_ww_main(B,VW1,VW2);
else
L = in_ww_main(B,VW1,0);
return append(L,[[V,DV,TMP_T,TMP_DT,Wt]]);
}
/*
returns [InData,AllData]
InData = [GIN,VDV,V,DV,WtV]
AllData = [G0,GIN0,VDV0,W,WVDV,WtV0]
GIN0 = ini_(-W,W)(G0)
WVDV = W[0]*V[0]*DV[0]+...
*/
def in_ww_main(F,VW1,VW2)
{
V = VW1[0]; DV = VW1[1]; WtV = VW1[2]; W = VW1[3];
dp_set_weight(WtV);
N = length(V);
N2 = N*2;
/* If W is a list, convert it to a vector */
if ( type(W) == 4 )
W = newvect(length(W),W);
dp_weyl_set_weight(W);
/* create a term order M in D<x,d> (DRL) */
M = newmat(N2,N2);
for ( J = 0; J < N2; J++ )
M[0][J] = 1;
for ( I = 1; I < N2; I++ )
M[I][N2-I] = -1;
VDV = append(V,DV);
/* create a non-term order MW in D<x,d> */
MW = newmat(N2+1,N2);
for ( J = 0; J < N; J++ )
MW[0][J] = -W[J];
for ( ; J < N2; J++ )
MW[0][J] = W[J-N];
for ( I = 1; I <= N2; I++ )
for ( J = 0; J < N2; J++ )
MW[I][J] = M[I-1][J];
/* create a homogenized term order MWH in D<x,d,h> */
MWH = newmat(N2+2,N2+1);
for ( J = 0; J <= N2; J++ )
MWH[0][J] = 1;
for ( I = 1; I <= N2+1; I++ )
for ( J = 0; J < N2; J++ )
MWH[I][J] = MW[I-1][J];
/* homogenize F */
VDVH = append(VDV,[TMP_H]);
FH = map(dp_dtop,map(dp_homo,map(dp_ptod,F,VDV)),VDVH);
/*
* FH is a GB w.r.t. any term order s.t. LT(FH)=[t,dx1,...,dxn]
* Compute a groebner basis of FH w.r.t. MWH by modular change of
* ordering.
* Since F is Z-coef, LC(FH)=[1,...,1] and we can use any prime p
* for trace algorithm.
*/
/* dp_gr_flags(["Top",1,"NoRA",1]); */
for ( I = 0, HC=[]; I <= N; I++ ) HC = cons(1,HC);
GH = nd_gb_candidate(FH,VDVH,11,0,HC,1);
/* dp_gr_flags(["Top",0,"NoRA",0]); */
/* dehomigenize GH */
G = map(subst,GH,TMP_H,1);
/* G is a groebner basis w.r.t. a non term order MW */
/* take the initial part w.r.t. (-W,W) */
GIN = map(initial_part,G,VDV,MW,W);
/* GIN is a groebner basis w.r.t. a term order M */
/* As -W+W=0, gr_(-W,W)(D<x,d>) = D<x,d> */
/* find b(W1*x1*d1+...+WN*xN*dN) in Id(GIN) */
for ( I = 0, T = 0; I < N; I++ )
T += W[I]*V[I]*DV[I];
AllData = [G,GIN,VDV,W,T,WtV];
if ( VW2 ) {
/* take LC(GIN) w.r.t. DRL */
dp_set_weight(WtV); dp_ord(0);
HC = map(dp_hc,map(dp_ptod,GIN,VDV));
V2 = VW2[0]; DV2 = VW2[1]; WtV2 = VW2[2];
VDV2 = append(V2,DV2);
dp_set_weight(WtV2);
GIN2 = nd_gb_candidate(GIN,VDV2,0,0,HC,1);
InData = [GIN2,VDV2,V2,DV2,WtV2];
} else {
if ( 0 ) {
dp_set_weight(WtV);
GIN1 = nd_weyl_gr_postproc(GIN,VDV,0,0,0);
InData = [GIN1,VDV,V,DV,WtV];
} else
InData = [GIN,VDV,V,DV,WtV];
}
/* B = weyl_minipoly(GIN2,VDV2,0,T); */ /* M represents DRL order */
WtV = dp_set_weight();
dp_set_weight(0);
return [InData,AllData];
}
/* annihilating ideal of F^s */
def ann(F)
{
if ( member(s,vars(F)) )
error("ann : the variable 's' is reserved.");
if ( type(Vord=getopt(vord)) == -1 ) Vord = 0;
F = ptozp(F);
V = vars(F);
if ( Vord ) {
Param = setminus(V,Vord);
V = Vord;
}
N = length(V);
D = newvect(N);
if ( type(Wt=getopt(weight)) == -1 )
for ( I = N-1, Wt = []; I >= 0; I-- ) Wt = append([V[I],1],Wt);
Wt1 = vector(N);
for ( I = 0, F1 =F; I < N; I++ ) {
VI = Wt[2*I]; WI = Wt[2*I+1];
for ( J = 0; J < N; J++ )
if ( VI == V[J] ) break;
F1 = subst(F1,VI,VI^WI);
}
for ( I = 0; I < N; I++ ) D[I] = [deg(F1,V[I]),V[I]];
qsort(D,ndbf.compare_first);
for ( V = [], I = 0; I < N; I++ ) V = cons(D[I][1],V);
V = reverse(V);
for ( I = 0; I < N; I++ ) {
VI = Wt[2*I]; WI = Wt[2*I+1];
for ( J = 0; J < N; J++ ) if ( VI == V[J] ) break;
Wt1[J] = WI;
}
Wt = vtol(Wt1);
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
W = append([TMP_Y1,TMP_Y2,TMP_T],V);
DW = append([TMP_DY1,TMP_DY2,TMP_DT],DV);
B = [1-TMP_Y1*TMP_Y2,TMP_T-TMP_Y1*F];
for ( I = 0; I < N; I++ ) {
B = cons(DV[I]+TMP_Y1*diff(F,V[I])*TMP_DT,B);
}
Tdeg = w_tdeg(F,V,Wt);
/* y1*y2-1, t-y1*f, dx1+y1*df/dx1*dt ... */
/* weight for [y1,y2,t, x1,...,xn, dy1,dy2, dt,dx1,...,dxn, h] */
/* 0 1 2 3 N3-1 N3 N3+1 N3+2 2*N3 */
/* 1 1 D+1 w1 wn 1 1 1 D D 1 */
N3 = N+3;
WtV = newvect(2*N3+1);
WtV[0] = WtV[1] = 1; WtV[2] = Tdeg+1;
for ( I = 3; I < N3; I++ ) WtV[I] = Wt[I-3];
for ( ; I <= N3+2; I++ ) WtV[I] = 1;
for ( ; I < 2*N3; I++ ) WtV[I] = Tdeg;
WtV[2*N3] = 1;
/* B is already a GB => modular change of ordering can be applied */
/* any prime is available => HC=[1] */
dp_set_weight(WtV);
G0 = nd_gb_candidate(B,append(W,DW),[[0,2],[0,length(W)*2-2]],0,[1],1);
dp_set_weight(0);
G1 = [];
for ( T = G0; T != []; T = cdr(T) ) {
E = car(T); VL = vars(E);
if ( !member(TMP_Y1,VL) && !member(TMP_Y2,VL) )
G1 = cons(E,G1);
}
G2 = map(psi,G1,TMP_T,TMP_DT);
G3 = map(subst,G2,TMP_T,-1-s);
return G3;
}
def in_gb_oaku(F)
{
if ( member(s,vars(F)) )
error("ann : the variable 's' is reserved.");
F = ptozp(F);
V = vars(F);
N = length(V);
D = newvect(N);
if ( type(Wt=getopt(weight)) == -1 )
for ( I = N-1, Wt = []; I >= 0; I-- ) Wt = append([V[I],1],Wt);
Wt1 = vector(N);
for ( I = 0, F1 =F; I < N; I++ ) {
VI = Wt[2*I]; WI = Wt[2*I+1];
for ( J = 0; J < N; J++ )
if ( VI == V[J] ) break;
F1 = subst(F1,VI,VI^WI);
}
for ( I = 0; I < N; I++ ) D[I] = [deg(F1,V[I]),V[I]];
qsort(D,ndbf.compare_first);
for ( V = [], I = 0; I < N; I++ ) V = cons(D[I][1],V);
V = reverse(V);
for ( I = 0; I < N; I++ ) {
VI = Wt[2*I]; WI = Wt[2*I+1];
for ( J = 0; J < N; J++ ) if ( VI == V[J] ) break;
Wt1[J] = WI;
}
Wt = vtol(Wt1);
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
W = append([TMP_Y1,TMP_Y2,TMP_T],V);
DW = append([TMP_DY1,TMP_DY2,TMP_DT],DV);
B = [TMP_T-TMP_Y1*F];
for ( I = 0; I < N; I++ ) {
B = cons(DV[I]+TMP_Y1*diff(F,V[I])*TMP_DT,B);
}
Tdeg = w_tdeg(F,V,Wt);
/* y1*y2-1, t-y1*f, dx1+y1*df/dx1*dt ... */
/* weight for [y1,y2,t, x1,...,xn, dy1,dy2, dt,dx1,...,dxn, h] */
/* 0 1 2 3 N3-1 N3 N3+1 N3+2 2*N3 */
/* 1 1 D+1 1 1 1 1 1 D D 1 */
N3 = N+3;
WtV = newvect(2*N3+1);
WtV[0] = WtV[1] = 1; WtV[2] = Tdeg+1;
for ( I = 3; I <= N3+2; I++ ) WtV[I] = 1;
for ( ; I < 2*N3; I++ ) WtV[I] = Tdeg;
WtV[2*N3] = 1;
/* B is already a GB => modular change of ordering can be applied */
/* any prime is available => HC=[1] */
dp_set_weight(WtV);
G0 = nd_gb_candidate(B,append(W,DW),[[0,2],[0,length(W)*2-2]],0,[1],1);
dp_set_weight(0);
G1 = map(subst,G0,TMP_Y1,1);
return [G1,append(V,DV)];
}
/* homogenization w.r.t. (-W,W)-weight */
/* VDV = [x1,...,xn,dx1,...,dxn] */
/* homogenize F w.r.t. (W,-W,1) for (x,dx,y) */
def homogenize_oaku(F,VDV,W,Y)
{
N = length(VDV);
if ( N%2 ) error("invalid variable list");
N2 = N/2;
if ( length(W) != N2 ) error("inconsistent weight vector");
W0 = dp_set_weight();
Wt = append(W,append(vtol(-ltov(W)),[1]));
dp_set_weight(Wt);
H = homogenize(F,VDV,Y);
dp_set_weight(W0);
if ( type(Vars=getopt(vars)) != -1 && Vars ) {
DY = strtov("d"+rtostr(Y));
for ( I = 0, T = VDV, V = []; I < N2; I++, T = cdr(T) )
V = cons(car(T),V);
T = cons(Y,append(T,[DY]));
for ( ; V != []; V = cdr(V) ) T = cons(car(V),T);
return [H,T];
} else return H;
}
/* F = [F0,F1,...] */
def ann_n(F)
{
L = length(F);
V = vars(F);
N = length(V);
D = newvect(N);
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
W = []; DW = [];
for ( I = L-1; I >= 0; I-- ) {
SI = rtostr(I);
W = cons(strtov("t"+SI),W);
DW = cons(strtov("dt"+SI),DW);
}
U = []; DU = [];
for ( I = L-1; I >= 0; I-- ) {
SI = rtostr(I);
U = append([strtov("u"+SI),strtov("v"+SI)],U);
DU = append([strtov("du"+SI),strtov("dv"+SI)],DU);
}
B = [];
for ( I = 0; I < N; I++ ) {
T = DV[I];
for ( J = 0; J < L; J++ )
T += U[2*J]*diff(F[J],V[I])*DW[J];
B = cons(T,B);
}
for ( I = 0; I < L; I++ )
B = append([W[I]-U[2*I]*F[I],1-U[2*I]*U[2*I+1]],B);
VA = append(U,append(W,V));
DVA = append(DU,append(DW,DV));
VDV = append(VA,DVA);
#if 0
G0 = nd_weyl_gr(B,VDV,0,[[0,2*L],[0,length(VDV)-2*L]]);
#else
G0 = nd_gb_candidate(B,VDV,[[0,2*L],[0,length(VDV)-2*L]],0,[1],1);
#endif
G1 = [];
for ( T = G0; T != []; T = cdr(T) ) {
E = car(T); VL = vars(E);
for ( TV = U; TV != []; TV = cdr(TV) )
if ( member(car(TV),VL) ) break;
if ( TV == [] )
G1 = cons(E,G1);
}
G2 = G1;
for ( I = 0; I < L; I++ ) {
G2 = map(psi,G2,W[I],DW[I]);
G2 = map(subst,G2,W[I],-1-strtov("s"+rtostr(I)));
}
return G2;
}
/*
* compute J_f|s=r, where r = the minimal integral root of global b_f(s)
* ann0(F) returns [MinRoot,Ideal]
*/
def ann0(F)
{
F = subst(F,s,TMP_S);
Ann = ann(F);
Bf = bfunction(F);
FList = cdr(fctr(Bf));
for ( T = FList, Min = 0; T != []; T = cdr(T) ) {
LF = car(car(T));
Root = -coef(LF,0)/coef(LF,1);
if ( dn(Root) == 1 && Root < Min )
Min = Root;
}
return [Min,map(subst,Ann,s,Min,TMP_S,s,TMP_DS,ds)];
}
/*
* For a polynomial F and a scalar A,
* compute generators of Ann(F^A).
*/
def ann_fa(F,A)
{
if ( type(Syz=getopt(syz)) == -1 ) Syz = 0;
F = subst(F,s,TMP_S);
Ann = ann(F);
Bf = bfunction(F);
FList = cdr(fctr(Bf));
for ( T = FList, Min = 0; T != []; T = cdr(T) ) {
LF = car(car(T));
Root = -coef(LF,0)/coef(LF,1);
if ( dn(Root) == 1 && Root < Min )
Min = Root;
}
D = A-Min;
if ( dn(D) != 1 || D <= 0 )
return map(ptozp,map(subst,Ann,s,A,TMP_S,s,TMP_DS,ds));
V = vars(F);
for ( I = length(V)-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
VDV = append(V,DV);
R = map(subst,Ann,s,Min,TMP_S,s,TMP_DS,ds);
F = ptozp(F);
if ( Syz ) {
/* syzygy method */
S = newsyz.module_syz(cons(F^D,R),VDV,1,0|weyl=1);
B = car(S);
for ( R = []; B != []; B = cdr(B) )
if ( H = car(car(B)) )
R = cons(ptozp(H),R);
} else {
/* colon method */
for ( I = 0; I < D; I++ )
R = weyl_ideal_quotient(R,F,VDV);
}
return R;
}
def psi0(F,T,DT)
{
D = dp_ptod(F,[T,DT]);
Wmax = ww_weight(D);
D1 = dp_rest(D);
for ( ; D1; D1 = dp_rest(D1) )
if ( ww_weight(D1) > Wmax )
Wmax = ww_weight(D1);
for ( D1 = D, Dmax = 0; D1; D1 = dp_rest(D1) )
if ( ww_weight(D1) == Wmax )
Dmax += dp_hm(D1);
if ( Wmax >= 0 )
Dmax = dp_weyl_mul(<<Wmax,0>>,Dmax);
else
Dmax = dp_weyl_mul(<<0,-Wmax>>,Dmax);
Rmax = dp_dtop(Dmax,[T,DT]);
return Rmax;
}
def psi(F,T,DT)
{
Rmax = psi0(F,T,DT);
R = b_subst(subst(Rmax,DT,1),T);
return R;
}
def ww_weight(D)
{
V = dp_etov(D);
return V[1]-V[0];
}
def compare_first(A,B)
{
A0 = car(A);
B0 = car(B);
if ( A0 > B0 )
return 1;
else if ( A0 < B0 )
return -1;
else
return 0;
}
/* generic b-function w.r.t. weight vector W */
def generic_bfct(F,V,DV,W)
{
N = length(V);
N2 = N*2;
/* If W is a list, convert it to a vector */
if ( type(W) == 4 )
W = newvect(length(W),W);
dp_weyl_set_weight(W);
/* create a term order M in D<x,d> (DRL) */
M = newmat(N2,N2);
for ( J = 0; J < N2; J++ )
M[0][J] = 1;
for ( I = 1; I < N2; I++ )
M[I][N2-I] = -1;
VDV = append(V,DV);
/* create a non-term order MW in D<x,d> */
MW = newmat(N2+1,N2);
for ( J = 0; J < N; J++ )
MW[0][J] = -W[J];
for ( ; J < N2; J++ )
MW[0][J] = W[J-N];
for ( I = 1; I <= N2; I++ )
for ( J = 0; J < N2; J++ )
MW[I][J] = M[I-1][J];
/* create a homogenized term order MWH in D<x,d,h> */
MWH = newmat(N2+2,N2+1);
for ( J = 0; J <= N2; J++ )
MWH[0][J] = 1;
for ( I = 1; I <= N2+1; I++ )
for ( J = 0; J < N2; J++ )
MWH[I][J] = MW[I-1][J];
/* homogenize F */
VDVH = append(VDV,[TMP_H]);
FH = map(dp_dtop,map(dp_homo,map(dp_ptod,F,VDV)),VDVH);
/* compute a groebner basis of FH w.r.t. MWH */
dp_gr_flags(["Top",1,"NoRA",1]);
GH = dp_weyl_gr_main(FH,VDVH,0,1,11);
dp_gr_flags(["Top",0,"NoRA",0]);
/* dehomigenize GH */
G = map(subst,GH,TMP_H,1);
/* G is a groebner basis w.r.t. a non term order MW */
/* take the initial part w.r.t. (-W,W) */
GIN = map(initial_part,G,VDV,MW,W);
/* GIN is a groebner basis w.r.t. a term order M */
/* As -W+W=0, gr_(-W,W)(D<x,d>) = D<x,d> */
/* find b(W1*x1*d1+...+WN*xN*dN) in Id(GIN) */
for ( I = 0, T = 0; I < N; I++ )
T += W[I]*V[I]*DV[I];
B = weyl_minipoly(GIN,VDV,0,T); /* M represents DRL order */
return B;
}
/* all term reduction + interreduce */
def generic_bfct_1(F,V,DV,W)
{
N = length(V);
N2 = N*2;
/* If W is a list, convert it to a vector */
if ( type(W) == 4 )
W = newvect(length(W),W);
dp_weyl_set_weight(W);
/* create a term order M in D<x,d> (DRL) */
M = newmat(N2,N2);
for ( J = 0; J < N2; J++ )
M[0][J] = 1;
for ( I = 1; I < N2; I++ )
M[I][N2-I] = -1;
VDV = append(V,DV);
/* create a non-term order MW in D<x,d> */
MW = newmat(N2+1,N2);
for ( J = 0; J < N; J++ )
MW[0][J] = -W[J];
for ( ; J < N2; J++ )
MW[0][J] = W[J-N];
for ( I = 1; I <= N2; I++ )
for ( J = 0; J < N2; J++ )
MW[I][J] = M[I-1][J];
/* create a homogenized term order MWH in D<x,d,h> */
MWH = newmat(N2+2,N2+1);
for ( J = 0; J <= N2; J++ )
MWH[0][J] = 1;
for ( I = 1; I <= N2+1; I++ )
for ( J = 0; J < N2; J++ )
MWH[I][J] = MW[I-1][J];
/* homogenize F */
VDVH = append(VDV,[TMP_H]);
FH = map(dp_dtop,map(dp_homo,map(dp_ptod,F,VDV)),VDVH);
/* compute a groebner basis of FH w.r.t. MWH */
/* dp_gr_flags(["Top",1,"NoRA",1]); */
GH = dp_weyl_gr_main(FH,VDVH,0,1,11);
/* dp_gr_flags(["Top",0,"NoRA",0]); */
/* dehomigenize GH */
G = map(subst,GH,TMP_H,1);
/* G is a groebner basis w.r.t. a non term order MW */
/* take the initial part w.r.t. (-W,W) */
GIN = map(initial_part,G,VDV,MW,W);
/* GIN is a groebner basis w.r.t. a term order M */
/* As -W+W=0, gr_(-W,W)(D<x,d>) = D<x,d> */
/* find b(W1*x1*d1+...+WN*xN*dN) in Id(GIN) */
for ( I = 0, T = 0; I < N; I++ )
T += W[I]*V[I]*DV[I];
B = weyl_minipoly(GIN,VDV,0,T); /* M represents DRL order */
return B;
}
def initial_part(F,V,MW,W)
{
N2 = length(V);
N = N2/2;
dp_ord(MW);
DF = dp_ptod(F,V);
R = dp_hm(DF);
DF = dp_rest(DF);
E = dp_etov(R);
for ( I = 0, TW = 0; I < N; I++ )
TW += W[I]*(-E[I]+E[N+I]);
RW = TW;
for ( ; DF; DF = dp_rest(DF) ) {
E = dp_etov(DF);
for ( I = 0, TW = 0; I < N; I++ )
TW += W[I]*(-E[I]+E[N+I]);
if ( TW == RW )
R += dp_hm(DF);
else if ( TW < RW )
break;
else
error("initial_part : cannot happen");
}
return dp_dtop(R,V);
}
/* b-function of F ? */
def bfct(F)
{
/* XXX */
F = replace_vars_f(F);
V = vars(F);
N = length(V);
D = newvect(N);
for ( I = 0; I < N; I++ )
D[I] = [deg(F,V[I]),V[I]];
qsort(D,ndbf.compare_first);
for ( V = [], I = 0; I < N; I++ )
V = cons(D[I][1],V);
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
V1 = cons(s,V); DV1 = cons(ds,DV);
G0 = indicial1(F,reverse(V));
G1 = dp_weyl_gr_main(G0,append(V1,DV1),0,1,0);
Minipoly = weyl_minipoly(G1,append(V1,DV1),0,s);
return Minipoly;
}
/* called from bfct() only */
def indicial1(F,V)
{
W = append([y1,t],V);
N = length(V);
B = [t-y1*F];
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
DW = append([dy1,dt],DV);
for ( I = 0; I < N; I++ ) {
B = cons(DV[I]+y1*diff(F,V[I])*dt,B);
}
dp_nelim(1);
/* homogenized (heuristics) */
G0 = dp_weyl_gr_main(B,append(W,DW),1,0,6);
G1 = map(subst,G0,y1,1);
G2 = map(psi,G1,t,dt);
G3 = map(subst,G2,t,-s-1);
return G3;
}
/* b-function computation via generic_bfct() (experimental) */
def bfct_via_gbfct(F)
{
V = vars(F);
N = length(V);
D = newvect(N);
for ( I = 0; I < N; I++ )
D[I] = [deg(F,V[I]),V[I]];
qsort(D,ndbf.compare_first);
for ( V = [], I = 0; I < N; I++ )
V = cons(D[I][1],V);
V = reverse(V);
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
B = [TMP_T-F];
for ( I = 0; I < N; I++ ) {
B = cons(DV[I]+diff(F,V[I])*TMP_DT,B);
}
V1 = cons(TMP_T,V); DV1 = cons(TMP_DT,DV);
W = newvect(N+1);
W[0] = 1;
R = generic_bfct(B,V1,DV1,W);
return subst(R,s,-s-1);
}
/* use an order s.t. [t,x,y,z,...,dt,dx,dy,dz,...,h] */
def bfct_via_gbfct_weight(F,V)
{
N = length(V);
D = newvect(N);
Wt = getopt(weight);
if ( type(Wt) != 4 ) {
for ( I = 0, Wt = []; I < N; I++ )
Wt = cons(1,Wt);
}
Tdeg = w_tdeg(F,V,Wt);
WtV = newvect(2*(N+1)+1);
WtV[0] = Tdeg;
WtV[N+1] = 1;
/* wdeg(V[I])=Wt[I], wdeg(DV[I])=Tdeg-Wt[I]+1 */
for ( I = 1; I <= N; I++ ) {
WtV[I] = Wt[I-1];
WtV[N+1+I] = Tdeg-Wt[I-1]+1;
}
WtV[2*(N+1)] = 1;
dp_set_weight(WtV);
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
B = [TMP_T-F];
for ( I = 0; I < N; I++ ) {
B = cons(DV[I]+diff(F,V[I])*TMP_DT,B);
}
V1 = cons(TMP_T,V); DV1 = cons(TMP_DT,DV);
W = newvect(N+1);
W[0] = 1;
R = generic_bfct_1(B,V1,DV1,W);
dp_set_weight(0);
return subst(R,s,-s-1);
}
/* use an order s.t. [x,y,z,...,t,dx,dy,dz,...,dt,h] */
def bfct_via_gbfct_weight_1(F,V)
{
N = length(V);
D = newvect(N);
Wt = getopt(weight);
if ( type(Wt) != 4 ) {
for ( I = 0, Wt = []; I < N; I++ )
Wt = cons(1,Wt);
}
Tdeg = w_tdeg(F,V,Wt);
WtV = newvect(2*(N+1)+1);
/* wdeg(V[I])=Wt[I], wdeg(DV[I])=Tdeg-Wt[I]+1 */
for ( I = 0; I < N; I++ ) {
WtV[I] = Wt[I];
WtV[N+1+I] = Tdeg-Wt[I]+1;
}
WtV[N] = Tdeg;
WtV[2*N+1] = 1;
WtV[2*(N+1)] = 1;
dp_set_weight(WtV);
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
B = [TMP_T-F];
for ( I = 0; I < N; I++ ) {
B = cons(DV[I]+diff(F,V[I])*TMP_DT,B);
}
V1 = append(V,[TMP_T]); DV1 = append(DV,[TMP_DT]);
W = newvect(N+1);
W[N] = 1;
R = generic_bfct_1(B,V1,DV1,W);
dp_set_weight(0);
return subst(R,s,-s-1);
}
def bfct_via_gbfct_weight_2(F,V)
{
N = length(V);
D = newvect(N);
Wt = getopt(weight);
if ( type(Wt) != 4 ) {
for ( I = 0, Wt = []; I < N; I++ )
Wt = cons(1,Wt);
}
Tdeg = w_tdeg(F,V,Wt);
/* a weight for the first GB computation */
/* [t,x1,...,xn,dt,dx1,...,dxn,h] */
WtV = newvect(2*(N+1)+1);
WtV[0] = Tdeg;
WtV[N+1] = 1;
WtV[2*(N+1)] = 1;
/* wdeg(V[I])=Wt[I], wdeg(DV[I])=Tdeg-Wt[I]+1 */
for ( I = 1; I <= N; I++ ) {
WtV[I] = Wt[I-1];
WtV[N+1+I] = Tdeg-Wt[I-1]+1;
}
dp_set_weight(WtV);
/* a weight for the second GB computation */
/* [x1,...,xn,t,dx1,...,dxn,dt,h] */
WtV2 = newvect(2*(N+1)+1);
WtV2[N] = Tdeg;
WtV2[2*N+1] = 1;
WtV2[2*(N+1)] = 1;
for ( I = 0; I < N; I++ ) {
WtV2[I] = Wt[I];
WtV2[N+1+I] = Tdeg-Wt[I]+1;
}
for ( I = N-1, DV = []; I >= 0; I-- )
DV = cons(strtov("d"+rtostr(V[I])),DV);
B = [TMP_T-F];
for ( I = 0; I < N; I++ ) {
B = cons(DV[I]+diff(F,V[I])*TMP_DT,B);
}
V1 = cons(TMP_T,V); DV1 = cons(TMP_DT,DV);
V2 = append(V,[TMP_T]); DV2 = append(DV,[TMP_DT]);
W = newvect(N+1,[1]);
dp_weyl_set_weight(W);
VDV = append(V1,DV1);
N1 = length(V1);
N2 = N1*2;
/* create a non-term order MW in D<x,d> */
MW = newmat(N2+1,N2);
for ( J = 0; J < N1; J++ ) {
MW[0][J] = -W[J]; MW[0][N1+J] = W[J];
}
for ( J = 0; J < N2; J++ ) MW[1][J] = 1;
for ( I = 2; I <= N2; I++ ) MW[I][N2-I+1] = -1;
/* homogenize F */
VDVH = append(VDV,[TMP_H]);
FH = map(dp_dtop,map(dp_homo,map(dp_ptod,B,VDV)),VDVH);
/* compute a groebner basis of FH w.r.t. MWH */
GH = dp_weyl_gr_main(FH,VDVH,0,1,11);
/* dehomigenize GH */
G = map(subst,GH,TMP_H,1);
/* G is a groebner basis w.r.t. a non term order MW */
/* take the initial part w.r.t. (-W,W) */
GIN = map(initial_part,G,VDV,MW,W);
/* GIN is a groebner basis w.r.t. a term order M */
/* As -W+W=0, gr_(-W,W)(D<x,d>) = D<x,d> */
/* find b(W1*x1*d1+...+WN*xN*dN) in Id(GIN) */
for ( I = 0, T = 0; I < N1; I++ )
T += W[I]*V1[I]*DV1[I];
/* change of ordering from VDV to VDV2 */
VDV2 = append(V2,DV2);
dp_set_weight(WtV2);
for ( Pind = 0; ; Pind++ ) {
Prime = lprime(Pind);
GIN2 = dp_weyl_gr_main(GIN,VDV2,0,-Prime,0);
if ( GIN2 ) break;
}
R = weyl_minipoly(GIN2,VDV2,0,T); /* M represents DRL order */
dp_set_weight(0);
return subst(R,s,-s-1);
}
/* minimal polynomial of s; modular computation */
def weyl_minipolym(G,V,O,M,V0)
{
N = length(V);
Len = length(G);
dp_ord(O);
setmod(M);
PS = newvect(Len);
PS0 = newvect(Len);
for ( I = 0, T = G; T != []; T = cdr(T), I++ )
PS0[I] = dp_ptod(car(T),V);
for ( I = 0, T = G; T != []; T = cdr(T), I++ )
PS[I] = dp_mod(dp_ptod(car(T),V),M,[]);
for ( I = Len - 1, GI = []; I >= 0; I-- )
GI = cons(I,GI);
U = dp_mod(dp_ptod(V0,V),M,[]);
U = dp_weyl_nf_mod(GI,U,PS,1,M);
T = dp_mod(<<0>>,M,[]);
TT = dp_mod(dp_ptod(1,V),M,[]);
G = H = [[TT,T]];
for ( I = 1; ; I++ ) {
if ( dp_gr_print() )
print(".",2);
T = dp_mod(<<I>>,M,[]);
TT = dp_weyl_nf_mod(GI,dp_weyl_mul_mod(TT,U,M),PS,1,M);
H = cons([TT,T],H);
L = dp_lnf_mod([TT,T],G,M);
if ( !L[0] ) {
if ( dp_gr_print() )
print("");
return dp_dtop(L[1],[t]); /* XXX */
} else
G = insert(G,L);
}
}
/* minimal polynomial of s over Q */
def weyl_minipoly(G0,V0,O0,P)
{
HM = hmlist(G0,V0,O0);
N = length(V0);
Len = length(G0);
dp_ord(O0);
PS = newvect(Len);
for ( I = 0, T = G0, HL = []; T != []; T = cdr(T), I++ )
PS[I] = dp_ptod(car(T),V0);
for ( I = Len - 1, GI = []; I >= 0; I-- )
GI = cons(I,GI);
PSM = newvect(Len);
DP = dp_ptod(P,V0);
for ( Pind = 0; ; Pind++ ) {
Prime = lprime(Pind);
if ( !valid_modulus(HM,Prime) )
continue;
setmod(Prime);
for ( I = 0, T = G0, HL = []; T != []; T = cdr(T), I++ )
PSM[I] = dp_mod(dp_ptod(car(T),V0),Prime,[]);
NFP = weyl_nf(GI,DP,1,PS);
NFPM = dp_mod(NFP[0],Prime,[])/ptomp(NFP[1],Prime);
NF = [[dp_ptod(1,V0),1]];
LCM = 1;
TM = dp_mod(<<0>>,Prime,[]);
TTM = dp_mod(dp_ptod(1,V0),Prime,[]);
GM = NFM = [[TTM,TM]];
for ( D = 1; ; D++ ) {
if ( dp_gr_print() )
print(".",2);
NFPrev = car(NF);
NFJ = weyl_nf(GI,
dp_weyl_mul(NFP[0],NFPrev[0]),NFP[1]*NFPrev[1],PS);
NFJ = remove_cont(NFJ);
NF = cons(NFJ,NF);
LCM = ilcm(LCM,NFJ[1]);
/* modular computation */
TM = dp_mod(<<D>>,Prime,[]);
TTM = dp_mod(NFJ[0],Prime,[])/ptomp(NFJ[1],Prime);
NFM = cons([TTM,TM],NFM);
LM = dp_lnf_mod([TTM,TM],GM,Prime);
if ( !LM[0] )
break;
else
GM = insert(GM,LM);
}
if ( dp_gr_print() )
print("");
U = NF[0][0]*idiv(LCM,NF[0][1]);
Coef = [];
for ( J = D-1; J >= 0; J-- ) {
Coef = cons(strtov("u"+rtostr(J)),Coef);
U += car(Coef)*NF[D-J][0]*idiv(LCM,NF[D-J][1]);
}
for ( UU = U, Eq = []; UU; UU = dp_rest(UU) )
Eq = cons(dp_hc(UU),Eq);
M = etom([Eq,Coef]);
B = henleq(M,Prime);
if ( dp_gr_print() )
print("");
if ( B ) {
R = 0;
for ( I = 0; I < D; I++ )
R += B[0][I]*s^I;
R += B[1]*s^D;
return R;
}
}
}
def weyl_nf(B,G,M,PS)
{
for ( D = 0; G; ) {
for ( U = 0, L = B; L != []; L = cdr(L) ) {
if ( dp_redble(G,R=PS[car(L)]) > 0 ) {
GCD = igcd(dp_hc(G),dp_hc(R));
CG = idiv(dp_hc(R),GCD); CR = idiv(dp_hc(G),GCD);
U = CG*G-dp_weyl_mul(CR*dp_subd(G,R),R);
if ( !U )
return [D,M];
D *= CG; M *= CG;
break;
}
}
if ( U )
G = U;
else {
D += dp_hm(G); G = dp_rest(G);
}
}
return [D,M];
}
def weyl_nf_quo_check(G,PS,R)
{
D = R[0]; M = R[1]; Coef = R[2];
Len = length(PS);
T = 0;
for ( I = 0; I < Len; I++ )
T += dp_weyl_mul(Coef[I],PS[I]);
return (M*G-T)==D;
}
def weyl_nf_quo(B,G,M,PS)
{
Len = length(PS);
Coef = vector(Len);
for ( D = 0; G; ) {
for ( U = 0, L = B; L != []; L = cdr(L) ) {
if ( dp_redble(G,R=PS[car(L)]) > 0 ) {
GCD = igcd(dp_hc(G),dp_hc(R));
CG = idiv(dp_hc(R),GCD); CR = idiv(dp_hc(G),GCD);
for ( I = 0; I < Len; I++ ) Coef[I] *= CG;
Q = CR*dp_subd(G,R);
Coef[car(L)] += Q;
U = CG*G-dp_weyl_mul(Q,R);
D *= CG; M *= CG;
if ( !U )
return [D,M,Coef];
break;
}
}
if ( U )
G = U;
else {
D += dp_hm(G); G = dp_rest(G);
}
}
return [D,M,Coef];
}
def weyl_nf_mod(B,G,PS,Mod)
{
for ( D = 0; G; ) {
for ( U = 0, L = B; L != []; L = cdr(L) ) {
if ( dp_redble(G,R=PS[car(L)]) > 0 ) {
CR = dp_hc(G)/dp_hc(R);
U = G-dp_weyl_mul_mod(CR*dp_mod(dp_subd(G,R),Mod,[]),R,Mod);
if ( !U )
return D;
break;
}
}
if ( U )
G = U;
else {
D += dp_hm(G); G = dp_rest(G);
}
}
return D;
}
def b_subst(F,V)
{
D = deg(F,V);
C = newvect(D+1);
for ( I = D; I >= 0; I-- )
C[I] = coef(F,I,V);
for ( I = 0, R = 0; I <= D; I++ )
if ( C[I] )
R += C[I]*v_factorial(V,I);
return R;
}
def v_factorial(V,N)
{
for ( J = N-1, R = 1; J >= 0; J-- )
R *= V-J;
return R;
}
def w_tdeg(F,V,W)
{
dp_set_weight(newvect(length(W),W));
T = dp_ptod(F,V);
for ( R = 0; T; T = cdr(T) ) {
D = dp_td(T);
if ( D > R ) R = D;
}
return R;
}
def replace_vars_f(F)
{
return subst(F,s,TMP_S,t,TMP_T,y1,TMP_Y1,y2,TMP_Y2);
}
def replace_vars_v(V)
{
V = replace_var(V,s,TMP_S);
V = replace_var(V,t,TMP_T);
V = replace_var(V,y1,TMP_Y1);
V = replace_var(V,y2,TMP_Y2);
return V;
}
def replace_var(V,X,Y)
{
V = reverse(V);
for ( R = []; V != []; V = cdr(V) ) {
Z = car(V);
if ( Z == X )
R = cons(Y,R);
else
R = cons(Z,R);
}
return R;
}
def action_on_gfs(P,V,GFS)
{
for ( T = V, DV = []; T != []; T = cdr(T) )
DV = cons(strtov("d"+rtostr(car(T))),DV);
V = append(append(V,[s]),reverse(cons(ds,DV)));
DP = dp_ptod(P,V);
N = length(V)/2;
for ( I = N-1, V0 = []; I >= 0; I-- )
V0 = cons(V[I],V0);
R = [];
for ( T = DP; T; T = dp_rest(T) )
R = cons(action_on_gfs_1(dp_hm(T),N,V0,GFS),R);
D = coef(car(R)[2],0);
for ( T = cdr(R); T != []; T = cdr(T) ) {
Di = coef(car(T)[2],0);
if ( Di < D )
D = Di;
}
F = GFS[1];
for ( T = R, G = 0; T != []; T = cdr(T) )
G += car(T)[0]*F^(car(T)[2]-(s+D));
while ( 1 ) {
G1 = tdiv(G,F);
if ( G1 ) {
G = G1;
D++;
} else
return [G,F,s+D];
}
}
def action_on_gfs_1(M,N,V,GFS)
{
G = GFS[0];
F = GFS[1];
S = GFS[2];
C = dp_hc(M);
E = dp_etov(M);
for ( I = 0; I < N; I++ ) {
VI = V[I];
C *= VI^E[I];
DFVI = diff(F,VI);
for ( J = 0, EI = E[I+N]; J < EI; J++, S-- )
G = diff(G,VI)*F+S*G*DFVI;
}
return [C*G,F,S];
}
/* stratification */
def weyl_subst(F,P,V)
{
VF = var(F);
D = deg(F,VF);
P = dp_ptod(P,V);
One = dp_ptod(1,V);
for ( R = 0, I = D; I >= 0; I-- )
R = dp_weyl_mul(R,P)+coef(F,I,VF)*One;
return dp_dtop(R,V);
}
def bfactor(F)
{
L=length(F);
for(I=0,B=1;I<L;I++)B*=F[I][0]^F[I][1];
return fctr(B);
}
def gen_a(K)
{
D = x^(K+1);
W = [];
for ( I = 1; I <= K; I++ ) {
D += (V=strtov("u"+rtostr(K-I+1)))*x^(K-I);
W = cons(I+1,cons(V,W));
}
F = res(x,D,diff(D,x));
return [D,F,reverse(W)];
}
def gen_d(K)
{
D = x^2*y+y^(K-1)+u1+u2*x+u3*x^2;
W = reverse([u1,2*K-2,u2,K,u3,2]);
U = [u3,u2,u1];
for ( I = 4; I <= K; I++ ) {
D += (V=strtov("u"+rtostr(I)))*y^(I-3);
W = cons((2*K-2)-2*(I-3),cons(V,W));
U = cons(V,U);
}
B = [D,diff(D,x),diff(D,y)];
G = nd_gr_trace(B,append([x,y],U),1,1,0);
G = nd_gr_trace(G,append([x,y],U),1,-1,[[0,2],[0,K]]);
E = elimination(G,U);
F = E[0];
return [D,F,reverse(W)];
}
def gen_dm(K)
{
D = x^2*y-y^(K-1)+u1+u2*x+u3*x^2;
W = reverse([u1,2*K-2,u2,K,u3,2]);
U = [u3,u2,u1];
for ( I = 4; I <= K; I++ ) {
D += (V=strtov("u"+rtostr(I)))*y^(I-3);
W = cons((2*K-2)-2*(I-3),cons(V,W));
U = cons(V,U);
}
B = [D,diff(D,x),diff(D,y)];
G = nd_gr_trace(B,append([x,y],U),1,1,0);
G = nd_gr_trace(G,append([x,y],U),1,-1,[[0,2],[0,K]]);
E = elimination(G,U);
F = E[0];
return [D,F,reverse(W)];
}
def elimination(G,V)
{
ANS=[];
NG=length(G);
for (I=NG-1;I>=0;I--)
{
VSet=vars(G[I]);
DIFF=setminus(VSet,V);
if ( DIFF ==[] )
{
ANS=cons(G[I],ANS);
}
}
return ANS;
}
def weyl_ideal_quotient(B,F,VDV)
{
T = ttttt; DT = dttttt;
J = cons((1-T)*F,vtol(ltov(B)*T));
N = length(VDV); N1 = N/2;
for ( I = N1-1, V1 = []; I >= 0; I-- )
V1 = cons(VDV[I],V1);
for ( I = 0, VDV1 = VDV; I < N1; I++ ) VDV1 = cdr(VDV1);
VDV1 = append(cons(T,V1),cons(DT,VDV1));
O1 = [[0,1],[0,N+1]];
GJ = nd_weyl_gr(J,VDV1,0,O1);
R = elimination(GJ,VDV);
return map(weyl_divide_by_right,R,F,VDV,0);
}
def bf_strat(F)
{
if ( member(s,vars(F)) )
error("ann : the variable 's' is reserved.");
dp_ord(0);
T0 = time();
if ( type(Heu=getopt(heuristic)) == -1 ) Heu = 0;
if ( type(Vord=getopt(vord)) == -1 || type(Vord) != 4 ) Vord = 0;
if ( type(Wt=getopt(weight)) == -1 ) Wt = 0;
if ( type(ElimIdeal=getopt(elimideal)) == -1 ) ElimIdeal = 0;
L = in_ww(F|weight=Wt,heuristic=Heu,vord=Vord);
T1 = time();
print(["in_ww",(T1[0]+T1[1])-(T0[0]+T0[1])]);
/* shortcuts */
V = vars(F);
dp_set_weight(0);
dp_ord(0);
Sing = sing(F,V);
if ( Sing[0] == 1 || Sing[0] == -1 ) {
return [[[F],[1],[[s+1,1]]],[[0],[F],[]]];
} else if ( zero_dim(Sing,V,0) ) {
N = length(V);
P0 = [];
for ( I = 0; I < N; I++ ) {
M = minipoly(Sing,V,0,V[I],TMP_S);
MF = cdr(fctr(M));
if ( length(MF) == 1 && deg(MF[0][0],TMP_S)==1 ) {
P0 = cons(subst(MF[0][0],TMP_S,V[I]),P0);
} else break;
}
if ( I == N ) {
Indata = L[0]; AllData = L[1]; VData = L[2];
GIN = Indata[0]; VDV = Indata[1]; WVDV = AllData[4];
W = Indata[4];
dp_set_weight(W);
B = weyl_minipoly(GIN,VDV,0,WVDV);
B = subst(B,s,-s-1);
dp_set_weight(0);
return [
[P0,[1],cdr(fctr(B))],
[[F],P0,[[s+1,1]]],
[[0],[F],[]]
];
}
}
L2 = bf_strat_stage2(L);
if ( ElimIdeal ) return L2;
S = bf_strat_stage3(L2);
R = [];
for ( T = S; T != []; T = cdr(T) ) {
Str = car(T);
B = Str[2];
B1 = [];
for ( U = B; U != []; U = cdr(U) )
B1 = cons([subst(car(U)[0],s,-s-1),car(U)[1]],B1);
B1 = reverse(B1);
R = cons([Str[0],Str[1],B1],R);
}
return reverse(R);
}
/* returns [QQ,V,V0,B,BF,W] */
/* QQ : ideal in C[x,s] (s=tdt), V=[x1,..,xn,t], V0 = [x1,..,xn] */
/* B : global b-function, BF : factor list of B, W : weight */
def bf_strat_stage2(L)
{
T0 = time();
InData = L[0]; VData = L[2];
G1 = InData[0]; VDV = InData[1]; W = InData[4]; W0 = VData[4];
N = length(VDV); N1 = N/2;
V = InData[2]; DV = InData[3];
T = VData[2]; DT = VData[3];
V0 = VData[0]; DVR = VData[1];
dp_set_weight(W);
for ( I = 0; DVR != []; I++, DVR = cdr(DVR) ) {
DVRV = cons(DT,append(cdr(DVR),V));
M = elim_mat(VDV,DVRV);
for ( K = 0; K < N; K++ )
M[1][K] = W[K];
dp_ord(0); D1 = map(dp_ptod,G1,VDV);
H1 = map(dp_ht,D1); HC1 = map(dp_hc,D1);
dp_ord(M); H2 = map(dp_ht,map(dp_ptod,G1,VDV));
if ( H1 == H2 )
G2 = G1;
else
G2 = nd_gb_candidate(G1,VDV,M,0,HC1,1);
G1 = elimination(G2,DVRV);
}
T1 = time();
B = weyl_minipoly(G1,VDV,0,T*DT);
T2 = time();
BF = cdr(fctr(B));
dp_set_weight(0);
G1 = map(psi0,G1,T,DT);
QQ = map(subst,map(b_subst,map(subst,G1,DT,1),T),T,var(B));
if ( type(getopt(ideal)) != -1 ) return [QQ,V];
print(["elim",(T1[0]+T1[1])-(T0[0]+T0[1])]);
print(["globalb",(T2[0]+T2[1])-(T1[0]+T1[1])]);
return [QQ,V,V0,B,BF,W0,DV];
}
def bf_strat_stage3(L)
{
T0 = time();
QQ = L[0]; V0 = L[2]; B = L[3]; BF = L[4]; W0 = L[5];
NF = length(BF);
Data = vector(NF);
W1 = W0? cons(1,append(W0,[1])) : 0;
for ( I = J = 0; I < NF; I++ ) {
DI = tower_in_p(QQ,B,BF[I],V0,W0);
NDI = length(DI);
dp_set_weight(W1);
for ( K = 0; K < J; K++ ) {
if ( length(DK=Data[K]) == NDI ) {
for ( L = 0; L < NDI; L++ ) {
CL = DI[L][1]; CK = DK[L][1];
if ( !zero_inclusion(CL,CK,V0)
|| !zero_inclusion(CK,CL,V0) ) break;
}
if ( L == NDI ) break;
}
}
dp_set_weight(0);
if ( K < J ) {
for ( L = 0, T = []; L < NDI; L++ ) {
#if 0
NewId = DK[L][1];
#else
NewId = ideal_intersection(DK[L][1],DI[L][1],V0,0);
#endif
T = cons([[DK[L][0][0]*DI[L][0][0],DK[L][0][1]],
NewId,DK[L][2]],T);
}
Data[K] = reverse(T);
} else
Data[J++] = DI;
}
Data1 = vector(J);
for ( I = 0; I < J; I++ )
Data1[I] = Data[I];
T1 = time();
Str = stratify_bf(Data1,V0,W0);
T2 = time();
print(["tower",(T1[0]+T1[1])-(T0[0]+T0[1])]);
print(["strat",(T2[0]+T2[1])-(T1[0]+T1[1])]);
return Str;
}
/*
InData = [GIN,VDV,V,DV,WtV]
AllData = [G0,GIN0,VDV0,W,WVDV,WtV0]
*/
def bf_local(F,P)
{
if ( member(s,vars(F)) )
error("ann : the variable 's' is reserved.");
/* F -> F/Fcont */
F1 = ptozp(F); Fcont = sdiv(F,F1); F = F1;
if ( type(Heu=getopt(heuristic)) == -1 ) Heu = 0;
if ( type(Vord=getopt(vord)) == -1 || type(Vord) != 4 ) Vord = 0;
if ( type(Wt=getopt(weight)) == -1 ) Wt = 0;
if ( type(Op=getopt(op)) == -1 ) Op = 0;
L = in_ww(F|weight=Wt,heuristic=Heu,vord=Vord);
InData = L[0]; AllData = L[1]; VData = L[2];
G = InData[0]; VDV = InData[1];
V = InData[2]; DV = InData[3];
V0 = VData[0]; DV0 = VData[1]; T = VData[2]; DT = VData[3]; W0 = VData[4];
L2 = bf_strat_stage2(L);
/* L2 = [QQ,V,V0,B,BF,W] */
/* QQ : ideal in C[x,s] (s=tdt), V=[t,x1,..,xn], V0 = [x1,..,xn] */
/* B : global b-function, BF : factor list of B, W : weight */
QQ = L2[0]; B = L2[3]; BF = L2[4]; W = L2[5];
NF = length(BF);
BP = [];
dp_set_weight(0);
for ( I = J = 0; I < NF; I++ ) {
List = compute_exponent(QQ,B,BF[I],P,V0,W0);
DI = List[0]; QQI = List[1];
if ( DI )
BP = cons([BF[I][0],DI],BP);
if ( I == 0 )
Id = QQI;
else
Id = ideal_intersection(Id,QQI,V0,0);
}
for ( List = Id; List != []; List = cdr(List) )
if ( subst_vars(car(List),P) )
break;
if ( List == [] ) error("bf_local : inconsitent intersection");
Ax = car(List);
LB = [];
for ( BPT = 1, List = BP; List != []; List = cdr(List) ) {
BPT *= car(List)[0]^car(List)[1];
LB = cons([subst(car(List)[0],s,-s-1),car(List)[1]],LB);
}
LB = reverse(LB);
if ( !Op ) return LB;
BPT = weyl_subst(BPT,T*DT,VDV);
/* computation using G0,GIN0,VDV0 */
G0 = AllData[0]; GIN0 = AllData[1]; VDV0 = AllData[2]; WtV0 = AllData[5];
dp_set_weight(WtV0); dp_ord(0);
PS = map(dp_ptod,GIN0,VDV0); Len = length(PS);
for ( I = Len-1, Ind = []; I >= 0; I-- ) Ind = cons(I,Ind);
/* QR = [D,M,Coef] */
AxBPT = dp_ptod(Ax*BPT,VDV0);
QR = weyl_nf_quo(Ind,AxBPT,1,PS);
if ( !weyl_nf_quo_check(AxBPT,PS,QR) ) error("bf_local : invalid quotient");
if ( QR[0] ) error("bf_local : invalid quotient");
Den = QR[1]; Coef = QR[2];
for ( I = 0, R = Den*AxBPT; I < Len; I++ )
R -= dp_weyl_mul(Coef[I],dp_ptod(G0[I],VDV0));
R = dp_dtop(R,VDV0);
CR = conv_tdt(R,F,V0,DV0,T,DT);
dp_set_weight(0);
Cont = cont(CR); CR /= Cont;
Cont *= dn(Fcont); Den *= nm(Fcont);
Gcd = igcd(Den,Cont);
return [LB,(Den/Gcd)*Ax,(Cont/Gcd)*CR];
}
/* t^(l+k)*dt^l (k>l) -> (s-k)(s-k-1)...(s-(k+l-1))t^k */
def conv_tdt(P,F,V0,DV0,T,DT)
{
DP = dp_ptod(P,[T,DT]);
VDV = append(cons(T,V0),cons(DT,DV0));
R = 0;
DF = dp_ptod(F,VDV);
for ( ; DP; DP = dp_rest(DP) ) {
C = dp_hc(DP);
E = dp_etov(dp_ht(DP));
L = E[1]; K = E[0]-E[1];
S = 1;
for ( I = 0; I < L; I++ )
S *= ((-T-1)-K-I);
U = dp_ptod(C*S,VDV);
for ( I = 1; I < K; I++ )
U = dp_weyl_mul(U,DF);
R += dp_dtop(U,VDV);
}
return subst(R,T,s);
}
/* W1=[W,1], W2=[1,W,1] */
def merge_tower(Str,Tower,V,W1,W2)
{
Prev = car(Tower); T = cdr(Tower);
Str1 = [];
for ( ; T != []; T = cdr(T) ) {
Cur = car(T);
Str1 = cons([Cur[1],Prev[1],[Prev[0]]],Str1);
Prev = Cur;
}
Str1 = cons([[0],Prev[1],[]],Str1);
Str1 = reverse(Str1);
if ( Str == [] ) return Str1;
U = [];
for ( T = Str; T != []; T = cdr(T) ) {
for ( S = Str1; S != []; S = cdr(S) ) {
Int = int_cons(T[0],S[0],V,W1,W2);
if ( Int[0] != [1] )
U = cons(append(Int,[append(T[0][2],S[0][2])]),U);
}
}
U = reverse(U);
return U;
}
def stratify_bf(Data,V,W)
{
N = length(Data);
R = [];
if ( W ) {
W1 = append(W,[1]);
W2 = cons(1,W1);
} else
W1 = W2 = 0;
for ( I = 0; I < N; I++ )
R = merge_tower(R,Data[I],V,W1,W2);
return R;
}
def tdt_homogenize(F,L)
{
TY1 = L[0]; T = TY1[0]; Y1 = TY1[1];
TY2 = L[1]; DT = TY2[0]; Y2 = TY2[1];
DF = dp_ptod(F,[T,DT]);
for ( R = 0; DF; DF = dp_rest(DF) ) {
M = dp_hm(DF);
E = dp_etov(M);
W = E[1]-E[0];
if ( W > 0 ) R += Y1^W*dp_dtop(M,[T,DT]);
else R += Y2^W*dp_dtop(M,[T,DT]);
}
return R;
}
def sing(F,V)
{
R = [F];
for ( T = V; T != []; T = cdr(T) )
R = cons(diff(F,car(T)),R);
G = nd_gr_trace(R,V,1,1,0);
return G;
}
def tower_in_p(B,F,FD,V,W)
{
TT = ttttt;
N = length(V); S = var(F); SV = cons(S,V); V1 = cons(TT,SV);
Wt = append(append([1,1],W),[1]);
dp_set_weight(Wt);
F1 = FD[0]; D = FD[1];
O1 = [[0,1],[0,N+1]]; O2 = [[0,1],[0,N]];
TF = sdiv(F,F1^D);
T = nd_gr_trace(cons((1-TT)*TF,vtol(TT*ltov(B))),V1,1,1,O1);
T = elimination(T,SV);
Q = map(sdiv,T,TF);
dp_set_weight(cdr(Wt));
QQ = elimination(nd_gr(Q,SV,0,O2),V);
E = [[[F1,0],QQ,PD]];
for ( I = D-1; I >= 0; I-- ) {
dp_set_weight(Wt);
T = nd_gr_trace(cons((1-TT)*F1,vtol(TT*ltov(Q))),V1,1,1,O1);
T = elimination(T,SV);
Q = map(sdiv,T,F1);
dp_set_weight(cdr(Wt));
QQ = elimination(nd_gr(Q,SV,0,O2),V);
E = cons([[F1,D-I],QQ,PD],E);
}
dp_set_weight(0);
return E;
}
def subst_vars(F,P)
{
for ( ; P != []; P = cdr(cdr(P)) )
F = subst(F,P[0],P[1]);
return F;
}
def compute_exponent(B,F,FD,P,V,W)
{
TT = ttttt;
N = length(V); S = var(F); SV = cons(S,V); V1 = cons(TT,SV);
F1 = FD[0]; D = FD[1];
Wt = append(append([1,1],W),[1]);
dp_set_weight(Wt);
O1 = [[0,1],[0,N+1]]; O2 = [[0,1],[0,N]];
TF = sdiv(F,F1^D);
T = nd_gr_trace(cons((1-TT)*TF,vtol(TT*ltov(B))),V1,0,1,O1);
T = elimination(T,SV);
Q = map(sdiv,T,TF);
QQ = elimination(nd_gr(Q,SV,0,O2),V);
for ( I = 0; I < D; I++ ) {
for ( T = QQ; T != []; T = cdr(T) )
if ( subst_vars(car(T),P) ) {
dp_set_weight(0);
return [I,QQ];
}
T = nd_gr_trace(cons((1-TT)*F1,vtol(TT*ltov(Q))),V1,0,1,O1);
T = elimination(T,SV);
Q = map(sdiv,T,F1);
QQ = elimination(nd_gr(Q,SV,0,O2),V);
}
dp_set_weight(0);
return [D,QQ];
}
/* V(B) subset V(A) ? */
def zero_inclusion(A,B,V)
{
NV = ttttt;
for ( T = A; T != []; T = cdr(T) ) {
F = car(T);
G = nd_gr_trace(cons(NV*F-1,B),cons(NV,V),1,1,0);
if ( type(car(G)) != 1 ) return 0;
}
return 1;
}
def weyl_divide_by_right(G,H,V,O)
{
dp_ord(O); G = dp_ptod(G,V); H = dp_ptod(H,V);
CH = dp_hc(H);
for ( Q = 0; G; ) {
if ( !dp_redble(G,H) ) return 0;
CG = dp_hc(G);
Q1 = CG/CH*dp_subd(G,H);
G -= dp_weyl_mul(Q1,H);
Q += Q1;
}
return dp_dtop(Q,V);
}
def elim_mat(V,W)
{
N = length(V);
M = matrix(N+1,N);
for ( J = 0; J < N; J++ ) if ( !member(V[J],W) ) M[0][J] = 1;
for ( J = 0; J < N; J++ ) M[1][J] = 1;
for ( I = 2; I <= N; I++ ) M[I][N-I+1] = -1;
return M;
}
/* (P-Q)cap(R-S)=(P cap Q^c)cap(R cap S^c)=(P cap R)cap(Q cup S)^c
=(P cap R)-(Q cup S)
*/
def int_cons(A,B,V,W1,W2)
{
AZ = A[0]; AN = A[1];
BZ = B[0]; BN = B[1];
if ( W1 ) dp_set_weight(W1);
CZ = nd_gr_trace(append(AZ,BZ),V,1,1,0);
if ( W2 ) dp_set_weight(W2);
CN = ideal_intersection(AN,BN,V,0);
ZI = zero_inclusion(CN,CZ,V);
dp_set_weight(0);
if ( ZI )
return [[1],[]];
else
return [CZ,CN];
}
def ideal_intersection(A,B,V,Ord)
{
T = ttttt;
G = nd_gr_trace(append(vtol(ltov(A)*T),vtol(ltov(B)*(1-T))),
cons(T,V),1,1,[[0,1],[Ord,length(V)]]);
return elimination(G,V);
}
def nd_gb_candidate(G,V,Ord,Homo,HC,Weyl)
{
Ind = 0;
N = length(HC);
while ( 1 ) {
P = lprime(Ind++);
for ( I = 0; I < N; I++ )
if ( !(HC[I]%P) ) break;
if ( I < N ) continue;
if ( Weyl )
G = nd_weyl_gr_trace(G,V,Homo,-P,Ord);
else
G = nd_gr_trace(G,V,Homo,-P,Ord);
if ( G ) return G;
}
}
endmodule $
end$
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