/* load["lib/all.k"];;
*/
/* $OpenXM: OpenXM/src/k097/lib/restriction/logc2.k,v 1.1 2007/11/28 01:39:53 takayama Exp $
The original was in misc-2007/11/logc2/logc2.kk
It is put under the cvs repository of openxm.org
*/
def Syz0(f) {
local ans;
sm1(" f syz /ans set ");
return(ans);
}
HelpAdd(["Syz0",
["Syz0 calls syz (sm1).",
"Example: Syz0([[\"z^2-1\",\"z-1\"], \"z\"]); "
]]);
def Syz0_xy(f) {
local ans;
sm1(" [(x,y) ring_of_differential_operators 0] define_ring f { . homogenize} map message ");
return( [1,1,1] );
}
/* Some test functions */
def logc2_pq(p,q) {
local f,ans;
RingD("x,y");
f = x^p+y^q+x*y^(q-1);
ans = Logc2(f);
return(ans);
}
/* cf. mail from Paco in Jan, 2007
logc2_pqab(4,7,1,1);
logc2_pqab(4,7,2,3); --> need minimal syzygy
*/
def logc2_pqab(p,q,a,b) {
local f,ans;
RingD("x,y");
f = (x^p+y^q+x*y^(q-1))*(x^a-y^b);
ans = Logc2(f);
return(ans);
}
HelpAdd(["Logc2",
["Logc2(f) [f a polynomial in x and y] computes dimensions",
"of the logarithmic cohomology groups.",
"load[\"lib/all.k\"];; is required to use it.",
"See Castro, Takayama: The Computation of the Logarithmic Cohomology for Plane Curves.",
"Example: Logc2(\"x*y*(x-y)\"): "
]]);
def Logc2(f) {
local s,ans,f,II,sss,pp,fx,fy;
sm1("0 set_timer "); sm1(" oxNoX ");
asssssir.OnTimer();
RingD("x,y");
/* f = x^p+y^q+x*y^(q-1); */
f = ReParse(f);
Print("f=");Println(f);
fx = Dx*f; fx = Replace(fx,[[Dx,Poly("0")],[h,Poly("1")]]);
fy = Dy*f; fy = Replace(fy,[[Dy,Poly("0")],[h,Poly("1")]]);
pp = [f,fx,fy];
Println(pp);
sss = Syz0([pp]);
sss = sss[0];
Println(sss);
if (Length(sss) != 2) Error("You need to use a function for Quillen-Suslin Theorem.");
p1 = -sss[0,0]+sss[0,1]*Dx+sss[0,2]*Dy;
p2 = -sss[1,0]+sss[1,1]*Dx+sss[1,2]*Dy;
SSS=sss;
Println([p1,p2]);
sm1(" [p1,p2] { [(x) (y) (Dx) (Dy) ] laplace0 } map /II set ");
Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
pp = Map(II,"ReParse");
Res = Sminimal(pp);
Res0 = Res[0];
Println("Step1: minimal resolution (Res0) "); sm1_pmat(Res0);
R = asir_BfRoots2(Res0[0]);
Println("Step2: computing the cohomology of the truncated complex.");
Print("Roots and b-function are "); Println(R);
R0 = R[0];
Ans=Srestall(Res0, ["x", "y"], ["x", "y"], R0[Length(R0)-1]);
Println("Timing data: sm1 "); sm1(" 1 set_timer ");
Print(" ox_asir [CPU,GC]: ");Println(asssssir.OffTimer());
Print("Answer is "); Println(Ans[0]);
return(Ans);
}