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version="version mregular.lib 4.0.0.0 Jun_2013 "; // $Id: 65c9ff28baa31013620d93b4f3033bcb6d5632ad $
category="Commutative Algebra";
info="
LIBRARY: mregular.lib Castelnuovo-Mumford regularity of homogeneous ideals
AUTHORS: I.Bermejo, ibermejo@ull.es
@* Ph.Gimenez, pgimenez@agt.uva.es
@* G.-M.Greuel, greuel@mathematik.uni-kl.de
OVERVIEW:
A library for computing the Castelnuovo-Mumford regularity of a homogeneous
ideal that DOES NOT require the computation of a minimal graded free
resolution of the ideal.
It also determines depth(basering/ideal) and satiety(ideal).
The procedures are based on 3 papers by Isabel Bermejo and Philippe Gimenez:
'On Castelnuovo-Mumford regularity of projective curves' Proc.Amer.Math.Soc.
128(5) (2000), 'Computing the Castelnuovo-Mumford regularity of some
subschemes of Pn using quotients of monomial ideals', Proceedings of
MEGA-2000, J. Pure Appl. Algebra 164 (2001), and 'Saturation and
Castelnuovo-Mumford regularity', Preprint (2004).
PROCEDURES:
regIdeal(id,[,e]); regularity of homogeneous ideal id
depthIdeal(id,[,e]); depth of S/id with S=basering, id homogeneous ideal
satiety(id,[,e]); saturation index of homogeneous ideal id
regMonCurve(li); regularity of projective monomial curve defined by li
NoetherPosition(id); Noether normalization of ideal id
is_NP(id); checks whether variables are in Noether position
is_nested(id); checks whether monomial ideal id is of nested type
";
LIB "general.lib";
LIB "algebra.lib";
LIB "sing.lib";
LIB "poly.lib";
//////////////////////////////////////////////////////////////////////////////
//
proc regIdeal (ideal i, list #)
"
USAGE: regIdeal (i[,e]); i ideal, e integer
RETURN: an integer, the Castelnuovo-Mumford regularity of i.
(returns -1 if i is not homogeneous)
ASSUME: i is a homogeneous ideal of the basering S=K[x(0)..x(n)].
e=0: (default)
If K is an infinite field, makes random changes of coordinates.
If K is a finite field, works over a transcendental extension.
e=1: Makes random changes of coordinates even when K is finite.
It works if it terminates, but may result in an infinite
loop. After 30 loops, a warning message is displayed and
-1 is returned.
NOTE: If printlevel > 0 (default = 0), additional info is displayed:
dim(S/i), depth(S/i) and end(H^(depth(S/i))(S/i)) are computed,
and an upper bound for the a-invariant of S/i is given.
The algorithm also determines whether the regularity is attained
or not at the last step of a minimal graded free resolution of i,
and if the answer is positive, the regularity of the Hilbert
function of S/i is given.
EXAMPLE: example regIdeal; shows some examples
"
{
//--------------------------- initialisation ---------------------------------
int e,ii,jj,H,h,d,time,lastv,sat,firstind;
int lastind,ch,nesttest,NPtest,nl,N,acc;
intmat ran;
def r0 = basering;
int n = nvars(r0)-1;
if ( size(#) > 0 )
{
e = #[1];
}
string s = "ring r1 = ",charstr(r0),",x(0..n),dp;";
execute(s);
ideal i,sbi,I,J,K,chcoord,m;
poly P;
map phi;
i = fetch(r0,i);
time=rtimer;
sbi=std(i);
ch=char(r1);
//----- Check ideal homogeneous
if ( homog(sbi) == 0 )
{
"// WARNING from proc regIdeal from lib mregular.lib:
// The ideal is not homogeneous!";
return (-1);
}
I=simplify(lead(sbi),1);
attrib(I,"isSB",1);
d=dim(I);
//----- If the ideal i is not proper:
if ( d == -1 )
{
dbprint(printlevel-voice+2,
"// The ideal i is (1)!
// Its Castelnuovo-Mumford regularity is:");
return (0);
}
//----- If the ideal i is 0:
if ( size(I) == 0 )
{
dbprint(printlevel-voice+2,
"// The ideal i is (0)!
// Its Castelnuovo-Mumford regularity is:");
return (0);
}
//----- When the ideal i is 0-dimensional:
if ( d == 0 )
{
H=maxdeg1(minbase(quotient(I,maxideal(1))))+1;
time=rtimer-time;
// Additional information:
dbprint(printlevel-voice+2,
"// Dimension of S/i : 0");
dbprint(printlevel-voice+2,
"// Time for computing regularity: " + string(time) + " sec.");
dbprint(printlevel-voice+2,
"// The Castelnuovo-Mumford regularity of i coincides with its satiety, and
// with the regularity of the Hilbert function of S/i. Its value is:");
return (H);
}
//----- Determine the situation: NT, or NP, or nothing.
//----- Choose the method depending on the situation, on the
//----- characteristic of the ground field, and on the option argument
//----- in order to get the mon. ideal of nested type associated to i
if ( e == 1 )
{ ch=0; }
NPtest=is_NP(I);
if ( NPtest == 1 )
{
nesttest=is_nested(I);
}
if ( ch != 0 )
{
if ( NPtest == 0 )
{
N=d*n-d*(d-1)/2;
s = "ring rtr = (ch,t(1..N)),x(0..n),dp;";
execute(s);
ideal chcoord,m,i,I;
poly P;
map phi;
i=imap(r1,i);
chcoord=select1(maxideal(1),1..(n-d+1));
acc=0;
for ( ii = 1; ii<=d; ii++ )
{
matrix trex[1][n-d+ii+1]=t((1+acc)..(n-d+ii+acc)),1;
m=select1(maxideal(1),1..(n-d+1+ii));
for ( jj = 1; jj<=n-d+ii+1; jj++ )
{
P=P+trex[1,jj]*m[jj];
}
chcoord[n-d+1+ii]=P;
P=0;
acc=acc+n-d+ii;
kill trex;
}
phi=rtr,chcoord;
I=simplify(lead(std(phi(i))),1);
setring r1;
I=imap(rtr,I);
attrib(I,"isSB",1);
}
else
{
if ( nesttest == 0 )
{
N=d*(d-1)/2;
s = "ring rtr = (ch,t(1..N)),x(0..n),dp;";
execute(s);
ideal chcoord,m,i,I;
poly P;
map phi;
i=imap(r1,i);
chcoord=select1(maxideal(1),1..(n-d+2));
acc=0;
for ( ii = 1; ii<=d-1; ii++ )
{
matrix trex[1][ii+1]=t((1+acc)..(ii+acc)),1;
m=select1(maxideal(1),(n-d+2)..(n-d+2+ii));
for ( jj = 1; jj<=ii+1; jj++ )
{
P=P+trex[1,jj]*m[jj];
}
chcoord[n-d+2+ii]=P;
P=0;
acc=acc+ii;
kill trex;
}
phi=rtr,chcoord;
I=simplify(lead(std(phi(i))),1);
setring r1;
I=imap(rtr,I);
attrib(I,"isSB",1);
}
}
}
else
{
if ( NPtest == 0 )
{
while ( nl < 30 )
{
chcoord=select1(maxideal(1),1..(n-d+1));
nl=nl+1;
for ( ii = 1; ii<=d; ii++ )
{
ran=random(100,1,n-d+ii);
ran=intmat(ran,1,n-d+ii+1);
ran[1,n-d+ii+1]=1;
m=select1(maxideal(1),1..(n-d+1+ii));
for ( jj = 1; jj<=n-d+ii+1; jj++ )
{
P=P+ran[1,jj]*m[jj];
}
chcoord[n-d+1+ii]=P;
P=0;
}
phi=r1,chcoord;
dbprint(printlevel-voice+2,"// (1 random change of coord.)");
I=simplify(lead(std(phi(i))),1);
attrib(I,"isSB",1);
NPtest=is_NP(I);
if ( NPtest == 1 )
{
break;
}
}
if ( NPtest == 0 )
{
"// WARNING from proc regIdeal from lib mregular.lib:
// The procedure has entered in 30 loops and could not put the variables
// in Noether position: in your example the method using random changes
// of coordinates may enter an infinite loop when the field is finite.
// Try removing this optional argument.";
return (-1);
}
i=phi(i);
nesttest=is_nested(I);
}
if ( nesttest == 0 )
{
while ( nl < 30 )
{
chcoord=select1(maxideal(1),1..(n-d+2));
nl=nl+1;
for ( ii = 1; ii<=d-1; ii++ )
{
ran=random(100,1,ii);
ran=intmat(ran,1,ii+1);
ran[1,ii+1]=1;
m=select1(maxideal(1),(n-d+2)..(n-d+2+ii));
for ( jj = 1; jj<=ii+1; jj++ )
{
P=P+ran[1,jj]*m[jj];
}
chcoord[n-d+2+ii]=P;
P=0;
}
phi=r1,chcoord;
dbprint(printlevel-voice+2,"// (1 random change of coord.)");
I=simplify(lead(std(phi(i))),1);
attrib(I,"isSB",1);
nesttest=is_nested(I);
if ( nesttest == 1 )
{
break;
}
}
if ( nesttest == 0 )
{
"// WARNING from proc regIdeal from lib mregular.lib:
// The procedure has entered in 30 loops and could not find a monomial
// ideal of nested type with the same regularity as your ideal: in your
// example the method using random changes of coordinates may enter an
// infinite loop when the field is finite.
// Try removing this optional argument.";
return (-1);
}
}
}
//
// At this stage, we have obtained a monomial ideal I of nested type
// such that reg(i)=reg(I). We now compute reg(I).
//
//----- When S/i is Cohen-Macaulay:
for ( ii = n-d+2; ii <= n+1; ii++ )
{
K=K+select(I,ii);
}
if ( size(K) == 0 )
{
s="ring nr = ",charstr(r0),",x(0..n-d),dp;";
execute(s);
ideal I;
I = imap(r1,I);
H=maxdeg1(minbase(quotient(I,maxideal(1))))+1;
time=rtimer-time;
// Additional information:
dbprint(printlevel-voice+2,
"// S/i is Cohen-Macaulay");
dbprint(printlevel-voice+2,
"// Dimension of S/i ( = depth(S/i) ): "+string(d));
dbprint(printlevel-voice+2,
"// Regularity attained at the last step of m.g.f.r. of i: YES");
dbprint(printlevel-voice+2,
"// Regularity of the Hilbert function of S/i: " + string(H-d));
dbprint(printlevel-voice+2,
"// Time for computing regularity: " + string(time) + " sec.");
dbprint(printlevel-voice+2,
"// The Castelnuovo-Mumford regularity of i is:");
return(H);
}
//----- When d=1:
if ( d == 1 )
{
H=maxdeg1(simplify(reduce(quotient(I,maxideal(1)),I),2))+1;
sat=H;
J=subst(I,x(n),1);
s = "ring nr = ",charstr(r0),",x(0..n-1),dp;";
execute(s);
ideal J=imap(r1,J);
attrib(J,"isSB",1);
h=maxdeg1(minbase(quotient(J,maxideal(1))))+1;
time=rtimer-time;
if ( h > H )
{
H=h;
}
// Additional information:
dbprint(printlevel-voice+2,
"// Dimension of S/i: 1");
dbprint(printlevel-voice+2,
"// Depth of S/i: 0");
dbprint(printlevel-voice+2,
"// Satiety of i: "+string(sat));
dbprint(printlevel-voice+2,
"// Upper bound for the a-invariant of S/i: end(H^1(S/i)) <= "+
string(h-2));
if ( H == sat )
{
dbprint(printlevel-voice+2,
"// Regularity attained at the last step of m.g.f.r. of i: YES");
dbprint(printlevel-voice+2,
"// Regularity of the Hilbert function of S/i: "+string(H));
}
else
{
dbprint(printlevel-voice+2,
"// Regularity attained at the last step of m.g.f.r. of i: NO");
}
dbprint(printlevel-voice+2,
"// Time for computing regularity: "+ string(time) + " sec.");
dbprint(printlevel-voice+2,
"// The Castelnuovo-Mumford regularity of i is:");
return(H);
}
//----- Now d>1 and S/i is not Cohen-Macaulay:
//
//----- First, determine the last variable really occuring
lastv=n-d;
h=n;
while ( lastv == n-d and h > n-d )
{
K=select(I,h+1);
if ( size(K) == 0 )
{
h=h-1;
}
else
{
lastv=h;
}
}
//----- and compute Castelnuovo-Mumford regularity:
s = "ring nr = ",charstr(r0),",x(0..lastv),dp;";
execute(s);
ideal I,K,KK,LL;
I=imap(r1,I);
attrib(I,"isSB",1);
K=simplify(reduce(quotient(I,maxideal(1)),I),2);
H=maxdeg1(K)+1;
firstind=H;
KK=minbase(subst(I,x(lastv),1));
for ( ii = n-lastv; ii<=d-2; ii++ )
{
LL=minbase(subst(I,x(n-ii-1),1));
attrib(LL,"isSB",1);
s = "ring mr = ",charstr(r0),",x(0..n-ii-1),dp;";
execute(s);
ideal K,KK;
KK=imap(nr,KK);
attrib(KK,"isSB",1);
K=simplify(reduce(quotient(KK,maxideal(1)),KK),2);
h=maxdeg1(K)+1;
if ( h > H )
{
H=h;
}
setring nr;
kill mr;
KK=LL;
}
// We must determine one more sat. index:
s = "ring mr = ",charstr(r0),",x(0..n-d),dp;";
execute(s);
ideal KK,K;
KK=imap(nr,KK);
attrib(KK,"isSB",1);
K=simplify(reduce(quotient(KK,maxideal(1)),KK),2);
h=maxdeg1(K)+1;
lastind=h;
if ( h > H )
{
H=h;
}
setring nr;
kill mr;
time=rtimer-time;
// Additional information:
dbprint(printlevel-voice+2,
"// Dimension of S/i: "+string(d));
dbprint(printlevel-voice+2,
"// Depth of S/i: "+string(n-lastv));
dbprint(printlevel-voice+2,
"// end(H^"+string(n-lastv)+"(S/i)) = "
+string(firstind-n+lastv-1));
dbprint(printlevel-voice+2,
"// Upper bound for the a-invariant of S/i: end(H^"
+string(d)+"(S/i)) <= "+string(lastind-d-1));
if ( H == firstind )
{
dbprint(printlevel-voice+2,
"// Regularity attained at the last step of m.g.f.r. of i: YES");
dbprint(printlevel-voice+2,
"// Regularity of the Hilbert function of S/i: "
+string(H-n+lastv));
}
else
{
dbprint(printlevel-voice+2,
"// Regularity attained at the last step of m.g.f.r. of i: NO");
}
dbprint(printlevel-voice+2,
"// Time for computing regularity: "+ string(time) + " sec.");
dbprint(printlevel-voice+2,
"// The Castelnuovo-Mumford regularity of i is:");
return(H);
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(x,y,z,t,w),dp;
ideal i=y2t,x2y-x2z+yt2,x2y2,xyztw,x3z2,y5+xz3w-x2zw2,x7-yt2w4;
regIdeal(i);
regIdeal(lead(std(i)));
// Additional information is displayed if you change printlevel (=1);
}
////////////////////////////////////////////////////////////////////////////////
/*
Out-commented examples:
//
ring s=0,x(0..5),dp;
ideal i=x(2)^2-x(4)*x(5),x(1)*x(2)-x(0)*x(5),x(0)*x(2)-x(1)*x(4),
x(1)^2-x(3)*x(5),x(0)*x(1)-x(2)*x(3),x(0)^2-x(3)*x(4);
regIdeal(i);
// Our procedure works when a min. graded free resol. can
// not be computed. In this easy example, regularity can also
// be obtained using a m.g.f.r.:
nrows(betti(mres(i,0)));
ring r1=0,(x,y,z,t),dp;
// Ex.2.5 in [Bermejo-Gimenez], Proc.Amer.Math.Soc. 128(5):
ideal i = x17y14-y31, x20y13, x60-y36z24-x20z20t20;
regIdeal(i);
// Ex.2.9 in [Bermejo-Gimenez], Proc.Amer.Math.Soc. 128(5):
int k=43;
ideal j=x17y14-y31,x20y13,x60-y36z24-x20z20t20,y41*z^k-y40*z^(k+1);
regIdeal(j);
k=14;
j=x17y14-y31,x20y13,x60-y36z24-x20z20t20,y41*z^k-y40*z^(k+1);
regIdeal(j);
k=22;
j=x17y14-y31,x20y13,x60-y36z24-x20z20t20,y41*z^k-y40*z^(k+1);
regIdeal(j);
k=315;
j=x17y14-y31,x20y13,x60-y36z24-x20z20t20,y41*z^k-y40*z^(k+1);
regIdeal(j);
// Example in Rk.2.10 in [Bermejo-Gimenez], ProcAMS 128(5):
ideal h=x2-3xy+5xt,xy-3y2+5yt,xz-3yz,2xt-yt,y2-yz-2yt;
regIdeal(h);
// The initial ideal is not saturated
regIdeal(lead(std(h)));
// More examples:
i=y4-t3z, x3t-y2z2, x3y2-t2z3, x6-tz5;
regIdeal(i);
//
regIdeal(maxideal(4));
//
ring r2=0,(x,y,z,t,w),dp;
ideal i = xy-zw,x3-yw2,x2z-y2w,y3-xz2,-y2z3+xw4+tw4+w5,-yz4+x2w3+xtw3+xw4,
-z5+x2tw2+x2w3+yw4;
regIdeal(i);
//
ring r3=0,(x,y,z,t,w,u),dp;
ideal i=imap(r2,i);
regIdeal(i);
// Next example is the defining ideal of the 2nd. Veronesean of P3, a variety
// in P8 which is arithmetically Cohen-Macaulay:
ring r4=0,(a,b,c,d,x(0..9)),dp;
ideal i= x(0)-ab,x(1)-ac,x(2)-ad,x(3)-bc,x(4)-bd,x(5)-cd,
x(6)-a2,x(7)-b2,x(8)-c2,x(9)-d2;
ideal ei=eliminate(i,abcd);
ring r5=0,x(0..9),dp;
ideal i=imap(r4,ei);
regIdeal(i);
// Here is an example where the computation of a m.g.f.r. of I costs:
ring r8=0,(x,y,z,t,u,a,b),dp;
ideal i=u-b40,t-a40,x-a23b17,y-a22b18+ab39,z-a25b15;
ideal ei=eliminate(i,ab); // It takes a few seconds to compute the ideal
ring r9=0,(x,y,z,t,u),dp;
ideal i=imap(r8,ei);
regIdeal(i); // This is very fast.
// Now you can use mres(i,0) to compute a m.g.f.r. of the ideal!
//
// The computation of the m.g.f.r. of the following example did not succeed
// using the command mres:
ring r10=0,(x(0..8),s,t),dp;
ideal i=x(0)-st24,x(1)-s2t23,x(2)-s3t22,x(3)-s9t16,x(4)-s11t14,x(5)-s18t7,
x(6)-s24t,x(7)-t25,x(8)-s25;
ideal ei=eliminate(i,st);
ring r11=0,x(0..8),dp;
ideal i=imap(r10,ei);
regIdeal(i);
// More examples where not even sres works:
// Be careful: elimination takes some time here, but it succeeds!
ring r12=0,(s,t,u,x(0..14)),dp;
ideal i=x(0)-st6u8,x(1)-s5t3u7,x(2)-t11u4,x(3)-s9t4u2,x(4)-s2t7u6,x(5)-s7t7u,
x(6)-s10t5,x(7)-s4t6u5,x(8)-s13tu,x(9)-s14u,x(10)-st2u12,x(11)-s3t9u3,
x(12)-s15,x(13)-t15,x(14)-u15;
ideal ei=eliminate(i,stu);
size(ei);
ring r13=0,x(0..14),dp;
ideal i=imap(r12,ei);
size(i);
regIdeal(i);
*/
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
proc depthIdeal (ideal i, list #)
"
USAGE: depthIdeal (i[,e]); i ideal, e integer
RETURN: an integer, the depth of S/i where S=K[x(0)..x(n)] is the basering.
(returns -1 if i is not homogeneous or if i=(1))
ASSUME: i is a proper homogeneous ideal.
e=0: (default)
If K is an infinite field, makes random changes of coordinates.
If K is a finite field, works over a transcendental extension.
e=1: Makes random changes of coordinates even when K is finite.
It works if it terminates, but may result in an infinite
loop. After 30 loops, a warning message is displayed and
-1 is returned.
NOTE: If printlevel > 0 (default = 0), dim(S/i) is also displayed.
EXAMPLE: example depthIdeal; shows some examples
"
{
//--------------------------- initialisation ---------------------------------
int e,ii,jj,h,d,time,lastv,ch,nesttest,NPtest,nl,N,acc;
intmat ran;
def r0 = basering;
int n = nvars(r0)-1;
if ( size(#) > 0 )
{
e = #[1];
}
string s = "ring r1 = ",charstr(r0),",x(0..n),dp;";
execute(s);
ideal i,sbi,I,J,K,chcoord,m;
poly P;
map phi;
i = fetch(r0,i);
time=rtimer;
sbi=std(i);
ch=char(r1);
//----- Check ideal homogeneous
if ( homog(sbi) == 0 )
{
"// WARNING from proc depthIdeal from lib mregular.lib:
// The ideal is not homogeneous!";
return (-1);
}
I=simplify(lead(sbi),1);
attrib(I,"isSB",1);
d=dim(I);
//----- If the ideal i is not proper:
if ( d == -1 )
{
"// WARNING from proc depthIdeal from lib mregular.lib:
// The ideal i is (1)!";
return (-1);
}
//----- If the ideal i is 0:
if ( size(I) == 0 )
{
dbprint(printlevel-voice+2,
"// The ideal i is (0)!
// The depth of S/i is:");
return (d);
}
//----- When the ideal i is 0-dimensional:
if ( d == 0 )
{
time=rtimer-time;
// Additional information:
dbprint(printlevel-voice+2,
"// Dimension of S/i : 0 (S/i is Cohen-Macaulay)");
dbprint(printlevel-voice+2,
"// Time for computing the depth: " + string(time) + " sec.");
dbprint(printlevel-voice+2,
"// The depth of S/i is:");
return (0);
}
//----- Determine the situation: NT, or NP, or nothing.
//----- Choose the method depending on the situation, on the
//----- characteristic of the ground field, and on the option argument
//----- in order to get the mon. ideal of nested type associated to i
if ( e == 1 )
{
ch=0;
}
NPtest=is_NP(I);
if ( NPtest == 1 )
{
nesttest=is_nested(I);
}
if ( ch != 0 )
{
if ( NPtest == 0 )
{
N=d*n-d*(d-1)/2;
s = "ring rtr = (ch,t(1..N)),x(0..n),dp;";
execute(s);
ideal chcoord,m,i,I;
poly P;
map phi;
i=imap(r1,i);
chcoord=select1(maxideal(1),1..(n-d+1));
acc=0;
for ( ii = 1; ii<=d; ii++ )
{
matrix trex[1][n-d+ii+1]=t((1+acc)..(n-d+ii+acc)),1;
m=select1(maxideal(1),1..(n-d+1+ii));
for ( jj = 1; jj<=n-d+ii+1; jj++ )
{
P=P+trex[1,jj]*m[jj];
}
chcoord[n-d+1+ii]=P;
P=0;
acc=acc+n-d+ii;
kill trex;
}
phi=rtr,chcoord;
I=simplify(lead(std(phi(i))),1);
setring r1;
I=imap(rtr,I);
attrib(I,"isSB",1);
}
else
{
if ( nesttest == 0 )
{
N=d*(d-1)/2;
s = "ring rtr = (ch,t(1..N)),x(0..n),dp;";
execute(s);
ideal chcoord,m,i,I;
poly P;
map phi;
i=imap(r1,i);
chcoord=select1(maxideal(1),1..(n-d+2));
acc=0;
for ( ii = 1; ii<=d-1; ii++ )
{
matrix trex[1][ii+1]=t((1+acc)..(ii+acc)),1;
m=select1(maxideal(1),(n-d+2)..(n-d+2+ii));
for ( jj = 1; jj<=ii+1; jj++ )
{
P=P+trex[1,jj]*m[jj];
}
chcoord[n-d+2+ii]=P;
P=0;
acc=acc+ii;
kill trex;
}
phi=rtr,chcoord;
I=simplify(lead(std(phi(i))),1);
setring r1;
I=imap(rtr,I);
attrib(I,"isSB",1);
}
}
}
else
{
if ( NPtest == 0 )
{
while ( nl < 30 )
{
chcoord=select1(maxideal(1),1..(n-d+1));
nl=nl+1;
for ( ii = 1; ii<=d; ii++ )
{
ran=random(100,1,n-d+ii);
ran=intmat(ran,1,n-d+ii+1);
ran[1,n-d+ii+1]=1;
m=select1(maxideal(1),1..(n-d+1+ii));
for ( jj = 1; jj<=n-d+ii+1; jj++ )
{
P=P+ran[1,jj]*m[jj];
}
chcoord[n-d+1+ii]=P;
P=0;
}
phi=r1,chcoord;
dbprint(printlevel-voice+2,"// (1 random change of coord.)");
I=simplify(lead(std(phi(i))),1);
attrib(I,"isSB",1);
NPtest=is_NP(I);
if ( NPtest == 1 )
{
break;
}
}
if ( NPtest == 0 )
{
"// WARNING from proc depthIdeal from lib mregular.lib:
// The procedure has entered in 30 loops and could not put the variables
// in Noether position: in your example the method using random changes
// of coordinates may enter an infinite loop when the field is finite.
// Try removing this optional argument.";
return (-1);
}
i=phi(i);
nesttest=is_nested(I);
}
if ( nesttest == 0 )
{
while ( nl < 30 )
{
chcoord=select1(maxideal(1),1..(n-d+2));
nl=nl+1;
for ( ii = 1; ii<=d-1; ii++ )
{
ran=random(100,1,ii);
ran=intmat(ran,1,ii+1);
ran[1,ii+1]=1;
m=select1(maxideal(1),(n-d+2)..(n-d+2+ii));
for ( jj = 1; jj<=ii+1; jj++ )
{
P=P+ran[1,jj]*m[jj];
}
chcoord[n-d+2+ii]=P;
P=0;
}
phi=r1,chcoord;
dbprint(printlevel-voice+2,"// (1 random change of coord.)");
I=simplify(lead(std(phi(i))),1);
attrib(I,"isSB",1);
nesttest=is_nested(I);
if ( nesttest == 1 )
{
break;
}
}
if ( nesttest == 0 )
{
"// WARNING from proc depthIdeal from lib mregular.lib:
// The procedure has entered in 30 loops and could not find a monomial
// ideal of nested type with the same depth as your ideal: in your
// example the method using random changes of coordinates may enter an
// infinite loop when the field is finite.
// Try removing this optional argument.";
return (-1);
}
}
}
//
// At this stage, we have obtained a monomial ideal I of nested type
// such that depth(S/i)=depth(S/I). We now compute depth(I).
//
//----- When S/i is Cohen-Macaulay:
for ( ii = n-d+2; ii <= n+1; ii++ )
{
K=K+select(I,ii);
}
if ( size(K) == 0 )
{
time=rtimer-time;
// Additional information:
dbprint(printlevel-voice+2,
"// Dimension of S/i: "+string(d)+" (S/i is Cohen-Macaulay)");
dbprint(printlevel-voice+2,
"// Time for computing depth: " + string(time) + " sec.");
dbprint(printlevel-voice+2,
"// The depth of S/i is:");
return(d);
}
//----- When d=1 (and S/i is not Cohen-Macaulay) ==> depth =0:
if ( d == 1 )
{
time=rtimer-time;
// Additional information:
dbprint(printlevel-voice+2,
"// Dimension of S/i: 1");
dbprint(printlevel-voice+2,
"// Time for computing depth: "+ string(time) + " sec.");
dbprint(printlevel-voice+2,
"// The depth of S/i is:");
return(0);
}
//----- Now d>1 and S/i is not Cohen-Macaulay:
//
//----- First, determine the last variable really occuring
lastv=n-d;
h=n;
while ( lastv == n-d and h > n-d )
{
K=select(I,h+1);
if ( size(K) == 0 )
{
h=h-1;
}
else
{
lastv=h;
}
}
//----- and compute the depth:
time=rtimer-time;
// Additional information:
dbprint(printlevel-voice+2,
"// Dimension of S/i: "+string(d));
dbprint(printlevel-voice+2,
"// Time for computing depth: "+ string(time) + " sec.");
dbprint(printlevel-voice+2,
"// The depth of S/i is:");
return(n-lastv);
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(x,y,z,t,w),dp;
ideal i=y2t,x2y-x2z+yt2,x2y2,xyztw,x3z2,y5+xz3w-x2zw2,x7-yt2w4;
depthIdeal(i);
depthIdeal(lead(std(i)));
// Additional information is displayed if you change printlevel (=1);
}
////////////////////////////////////////////////////////////////////////////////
/*
Out-commented examples:
ring s=0,x(0..5),dp;
ideal i=x(2)^2-x(4)*x(5),x(1)*x(2)-x(0)*x(5),x(0)*x(2)-x(1)*x(4),
x(1)^2-x(3)*x(5),x(0)*x(1)-x(2)*x(3),x(0)^2-x(3)*x(4);
depthIdeal(i);
// Our procedure works when a min. graded free resol. can
// not be computed. In this easy example, depth can also
// be obtained using a m.g.f.r. (Auslander-Buchsbaum formula):
nvars(s)-ncols(betti(mres(i,0)))+1;
ring r1=0,(x,y,z,t),dp;
// Ex.2.5 in [Bermejo-Gimenez], Proc.Amer.Math.Soc. 128(5):
ideal i = x17y14-y31, x20y13, x60-y36z24-x20z20t20;
depthIdeal(i);
// Ex.2.9 in [Bermejo-Gimenez], Proc.Amer.Math.Soc. 128(5):
int k=43;
ideal j=x17y14-y31,x20y13,x60-y36z24-x20z20t20,y41*z^k-y40*z^(k+1);
depthIdeal(j);
// Example in Rk.2.10 in [Bermejo-Gimenez], ProcAMS 128(5):
ideal h=x2-3xy+5xt,xy-3y2+5yt,xz-3yz,2xt-yt,y2-yz-2yt;
depthIdeal(h);
// The initial ideal is not saturated
depthIdeal(lead(std(h)));
// More examples:
i=y4-t3z, x3t-y2z2, x3y2-t2z3, x6-tz5;
depthIdeal(i);
//
depthIdeal(maxideal(4));
//
ring r2=0,(x,y,z,t,w),dp;
ideal i = xy-zw,x3-yw2,x2z-y2w,y3-xz2,-y2z3+xw4+tw4+w5,-yz4+x2w3+xtw3+xw4,
-z5+x2tw2+x2w3+yw4;
depthIdeal(i);
//
ring r3=0,(x,y,z,t,w,u),dp;
ideal i=imap(r2,i);
depthIdeal(i);
// Next example is the defining ideal of the 2nd. Veronesean of P3, a variety
// in P8 which is arithmetically Cohen-Macaulay:
ring r4=0,(a,b,c,d,x(0..9)),dp;
ideal i= x(0)-ab,x(1)-ac,x(2)-ad,x(3)-bc,x(4)-bd,x(5)-cd,
x(6)-a2,x(7)-b2,x(8)-c2,x(9)-d2;
ideal ei=eliminate(i,abcd);
ring r5=0,x(0..9),dp;
ideal i=imap(r4,ei);
depthIdeal(i);
// Here is an example where the computation of a m.g.f.r. of I costs:
ring r8=0,(x,y,z,t,u,a,b),dp;
ideal i=u-b40,t-a40,x-a23b17,y-a22b18+ab39,z-a25b15;
ideal ei=eliminate(i,ab); // It takes a few seconds to compute the ideal
ring r9=0,(x,y,z,t,u),dp;
ideal i=imap(r8,ei);
depthIdeal(i); // This is very fast.
// Now you can use mres(i,0) to compute a m.g.f.r. of the ideal!
//
// Another one:
ring r10=0,(x(0..8),s,t),dp;
ideal i=x(0)-st24,x(1)-s2t23,x(2)-s3t22,x(3)-s9t16,x(4)-s11t14,x(5)-s18t7,
x(6)-s24t,x(7)-t25,x(8)-s25;
ideal ei=eliminate(i,st);
ring r11=0,x(0..8),dp;
ideal i=imap(r10,ei);
depthIdeal(i);
// More examples where not even sres works:
// Be careful: elimination takes some time here, but it succeeds!
ring r12=0,(s,t,u,x(0..14)),dp;
ideal i=x(0)-st6u8,x(1)-s5t3u7,x(2)-t11u4,x(3)-s9t4u2,x(4)-s2t7u6,x(5)-s7t7u,
x(6)-s10t5,x(7)-s4t6u5,x(8)-s13tu,x(9)-s14u,x(10)-st2u12,x(11)-s3t9u3,
x(12)-s15,x(13)-t15,x(14)-u15;
ideal ei=eliminate(i,stu);
size(ei);
ring r13=0,x(0..14),dp;
ideal i=imap(r12,ei);
size(i);
depthIdeal(i);
//
*/
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
proc satiety (ideal i, list #)
"
USAGE: satiety (i[,e]); i ideal, e integer
RETURN: an integer, the satiety of i.
(returns -1 if i is not homogeneous)
ASSUME: i is a homogeneous ideal of the basering S=K[x(0)..x(n)].
e=0: (default)
The satiety is computed determining the fresh elements in the
socle of i. It works over arbitrary fields.
e=1: Makes random changes of coordinates to find a monomial ideal
with same satiety. It works over infinite fields only. If K
is finite, it works if it terminates, but may result in an
infinite loop. After 30 loops, a warning message is displayed
and -1 is returned.
THEORY: The satiety, or saturation index, of a homogeneous ideal i is the
least integer s such that, for all d>=s, the degree d part of the
ideals i and isat=sat(i,maxideal(1))[1] coincide.
NOTE: If printlevel > 0 (default = 0), dim(S/i) is also displayed.
EXAMPLE: example satiety; shows some examples
"
{
//--------------------------- initialisation ---------------------------------
int e,ii,jj,h,d,time,lastv,nesttest,NPtest,nl,sat;
intmat ran;
def r0 = basering;
int n = nvars(r0)-1;
if ( size(#) > 0 )
{
e = #[1];
}
string s = "ring r1 = ",charstr(r0),",x(0..n),dp;";
execute(s);
ideal i,sbi,I,K,chcoord,m,KK;
poly P;
map phi;
i = fetch(r0,i);
time=rtimer;
sbi=std(i);
//----- Check ideal homogeneous
if ( homog(sbi) == 0 )
{
"// WARNING from proc satiety from lib mregular.lib:
// The ideal is not homogeneous!";
return (-1);
}
I=simplify(lead(sbi),1);
attrib(I,"isSB",1);
d=dim(I);
//----- If the ideal i is not proper:
if ( d == -1 )
{
dbprint(printlevel-voice+2,
"// The ideal i is (1)!
// Its satiety is:");
return (0);
}
//----- If the ideal i is 0:
if ( size(I) == 0 )
{
dbprint(printlevel-voice+2,
"// The ideal i is (0)!
// Its satiety is:");
return (0);
}
//----- When the ideal i is 0-dimensional:
if ( d == 0 )
{
sat=maxdeg1(minbase(quotient(I,maxideal(1))))+1;
time=rtimer-time;
// Additional information:
dbprint(printlevel-voice+2,
"// Dimension of S/i: 0");
dbprint(printlevel-voice+2,
"// Time for computing the satiety: " + string(time) + " sec.");
dbprint(printlevel-voice+2,
"// The satiety of i is:");
return (sat);
}
//----- When one has option e=1:
//
//----- Determine the situation: NT, or NP, or nothing.
//----- Choose the method depending on the situation in order to
//----- get the mon. ideal of nested type associated to i
if ( e == 1 )
{
NPtest=is_NP(I);
if ( NPtest == 0 )
{
while ( nl < 30 )
{
chcoord=select1(maxideal(1),1..(n-d+1));
nl=nl+1;
for ( ii = 1; ii<=d; ii++ )
{
ran=random(100,1,n-d+ii);
ran=intmat(ran,1,n-d+ii+1);
ran[1,n-d+ii+1]=1;
m=select1(maxideal(1),1..(n-d+1+ii));
for ( jj = 1; jj<=n-d+ii+1; jj++ )
{
P=P+ran[1,jj]*m[jj];
}
chcoord[n-d+1+ii]=P;
P=0;
}
phi=r1,chcoord;
dbprint(printlevel-voice+2,"// (1 random change of coord.)");
I=simplify(lead(std(phi(i))),1);
attrib(I,"isSB",1);
NPtest=is_NP(I);
if ( NPtest == 1 )
{
break;
}
}
if ( NPtest == 0 )
{
"// WARNING from proc satiety from lib mregular.lib:
// The procedure has entered in 30 loops and could not put the variables
// in Noether position: in your example the method using random changes
// of coordinates may enter an infinite loop when the field is finite.
// Try removing the optional argument.";
return (-1);
}
i=phi(i);
}
nesttest=is_nested(I);
if ( nesttest == 0 )
{
while ( nl < 30 )
{
chcoord=select1(maxideal(1),1..(n-d+2));
nl=nl+1;
for ( ii = 1; ii<=d-1; ii++ )
{
ran=random(100,1,ii);
ran=intmat(ran,1,ii+1);
ran[1,ii+1]=1;
m=select1(maxideal(1),(n-d+2)..(n-d+2+ii));
for ( jj = 1; jj<=ii+1; jj++ )
{
P=P+ran[1,jj]*m[jj];
}
chcoord[n-d+2+ii]=P;
P=0;
}
phi=r1,chcoord;
dbprint(printlevel-voice+2,"// (1 random change of coord.)");
I=simplify(lead(std(phi(i))),1);
attrib(I,"isSB",1);
nesttest=is_nested(I);
if ( nesttest == 1 )
{
break;
}
}
if ( nesttest == 0 )
{
"// WARNING from proc satiety from lib mregular.lib:
// The procedure has entered in 30 loops and could not find a monomial
// ideal of nested type with the same satiety as your ideal: in your
// example the method using random changes of coordinates may enter an
// infinite loop when the field is finite.
// Try removing the optional argument.";
return (-1);
}
}
//
// At this stage, we have obtained a monomial ideal I of nested type
// such that depth(S/i)=depth(S/I). We now compute depth(I).
//
//----- When S/i is Cohen-Macaulay:
//
for ( ii = n-d+2; ii <= n+1; ii++ )
{
K=K+select(I,ii);
}
if ( size(K) == 0 )
{
time=rtimer-time;
// Additional information:
dbprint(printlevel-voice+2,
"// Dimension of S/i: "+string(d));
dbprint(printlevel-voice+2,
"// Time for computing satiety: " + string(time) + " sec.");
dbprint(printlevel-voice+2,
"// The satiety of i is:");
return(0);
}
//----- When d=1 (and S/i is not Cohen-Macaulay) ==> depth =0:
if ( d == 1 )
{
KK=simplify(reduce(quotient(I,maxideal(1)),I),2);
sat=maxdeg1(KK)+1;
time=rtimer-time;
// Additional information:
dbprint(printlevel-voice+2,
"// Dimension of S/i: 1");
dbprint(printlevel-voice+2,
"// Time for computing satiety: "+ string(time) + " sec.");
dbprint(printlevel-voice+2,
"// The satiety of i is:");
return(sat);
}
//----- Now d>1 and S/i is not Cohen-Macaulay:
//
//----- First, determine the last variable really occuring
lastv=n-d;
h=n;
while ( lastv == n-d and h > n-d )
{
K=select(I,h+1);
if ( size(K) == 0 )
{
h=h-1;
}
else
{
lastv=h;
}
}
//----- and compute the satiety:
sat=0;
if ( lastv == n )
{
KK=simplify(reduce(quotient(I,maxideal(1)),I),2);
sat=maxdeg1(KK)+1;
}
time=rtimer-time;
// Additional information:
dbprint(printlevel-voice+2,
"// Dimension of S/i: "+string(d));
dbprint(printlevel-voice+2,
"// Time for computing satiety: "+ string(time) + " sec.");
dbprint(printlevel-voice+2,
"// The satiety of i is:");
return(sat);
}
//---- If no option: direct computation
sat=maxdeg1(reduce(quotient(i,maxideal(1)),sbi))+1;
time=rtimer-time;
// Additional information:
dbprint(printlevel-voice+2,
"// Dimension of S/i: "+string(d)+";");
dbprint(printlevel-voice+2,
"// Time for computing satiety: "+ string(time) + " sec.");
dbprint(printlevel-voice+2,
"// The satiety of i is:");
return(sat);
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(x,y,z,t,w),dp;
ideal i=y2t,x2y-x2z+yt2,x2y2,xyztw,x3z2,y5+xz3w-x2zw2,x7-yt2w4;
satiety(i);
ideal I=lead(std(i));
satiety(I); // First method: direct computation
satiety(I,1); // Second method: doing changes of coordinates
// Additional information is displayed if you change printlevel (=1);
}
////////////////////////////////////////////////////////////////////////////////
/*
Out-commented examples:
ring s1=0,(x,y,z,t),dp;
ideal I=zt3,z2t2,yz2t,xz2t,xy2t,x3y;
satiety(I);
satiety(I,1);
// Another example:
ring s2=0,(z,y,x),dp;
ideal I=z38,z26y2,z14y4,z12x,z10x5,z8x9,z6x16,z4x23,z2y6,y32;
satiety(I);
satiety(I,1);
// One more:
ring s3=0,(s,t,u,x(0..8)),dp;
ideal i=x(0)-st6u8,x(1)-s5t3u7,x(2)-t11u4,x(3)-s9t4u2,
x(4)-s2t7u6,x(5)-s7t7u,x(6)-s15,x(7)-t15,x(8)-u15;
ideal ei=eliminate(i,stu);
size(ei);
ring s4=0,x(0..8),dp;
ideal i=imap(s3,ei);
ideal m=maxideal(1);
m[8]=m[8]+m[7];
map phi=m;
ideal phii=phi(i);
ideal nI=lead(std(phii));
ring s5=0,x(0..7),dp;
ideal nI=imap(s4,nI);
satiety(nI);
satiety(nI,1);
ideal I1=subst(nI,x(7),1);
ring s6=0,x(0..6),dp;
ideal I1=imap(s5,I1);
satiety(I1);
satiety(I1,1);
ideal I2=subst(I1,x(6),1);
ring s7=0,x(0..5),dp;
ideal I2=imap(s6,I2);
satiety(I2);
satiety(I2,1);
//
*/
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
proc regMonCurve (list #)
"
USAGE: regMonCurve (a0,...,an) ; ai integers with a0=0 < a1 < ... < an=:d
RETURN: an integer, the Castelnuovo-Mumford regularity of the projective
monomial curve C in Pn(K) parametrically defined by
x(0) = t^d , x(1) = s^(a1)t^(d-a1) , ..... , x(n) = s^d
where K is the field of complex numbers.
(returns -1 if a0=0 < a1 < ... < an is not satisfied)
ASSUME: a0=0 < a1 < ... < an are integers.
NOTES: 1. The defining ideal of the curve C, I in S=K[x(0),...,x(n)], is
determined by elimination.
2. The procedure regIdeal has been improved in this case since one
knows beforehand that the monomial ideal J=lead(std(I)) is of
nested type if the monomial ordering is dp, and that
reg(C)=reg(J) (see preprint 'Saturation and Castelnuovo-Mumford
regularity' by Bermejo-Gimenez, 2004).
3. If printlevel > 0 (default = 0) additional info is displayed:
- It says whether C is arithmetically Cohen-Macaulay or not.
- If C is not arith. Cohen-Macaulay, end(H^1(S/I)) is computed
and an upper bound for the a-invariant of S/I is given.
- It also determines one step of the minimal graded free
resolution (m.g.f.r.) of I where the regularity is attained
and gives the value of the regularity of the Hilbert function
of S/I when reg(I) is attained at the last step of a m.g.f.r.
EXAMPLE: example regMonCurve; shows some examples
"
{
//--------------------------- initialisation ---------------------------------
int ii,H,h,hh,time,ttime,firstind,lastind;
int n = size(#)-1;
//------------------ Check assumptions on integers -------------------------
if ( #[1] != 0 )
{"// WARNING from proc regMonCurve from lib mregular.lib:
// USAGE: your input must be a list of integers a0,a1,...,an such that
// a0=0 < a1 < a2 < ... < an";
return(-1);
}
for ( ii=1; ii<= n; ii++ )
{
if ( #[ii] >= #[ii+1] )
{
"// WARNING from proc regMonCurve from lib mregular.lib:
// USAGE: your input must be a list of integers a0,a1,...,an such that
// a0=0 < a1 < a2 < ... < an";
return(-1);
}
}
ring R=0,(x(0..n),s,t),dp;
ideal param,m,i;
poly f(0..n);
for (ii=0;ii<=n;ii++)
{
f(ii)=s^(#[n+1]-#[ii+1])*t^(#[ii+1]);
param=param+f(ii);
}
m=subst(maxideal(1),s,0);
m=simplify(subst(m,t,0),2);
i=matrix(m)-matrix(param);
ttime=rtimer;
i=eliminate(i,st);
ring r=0,(x(1..n),x(0)),dp;
ideal i,I;
i=imap(R,i);
I=minbase(lead(std(i)));
attrib(I,"isSB",1);
ttime=rtimer-ttime;
time=rtimer;
ring nr=0,x(1..n),dp;
ideal I,K,KK,J;
I=imap(r,I);
attrib(I,"isSB",1);
K=select(I,n);
//------------------ Cohen-Macaulay case ------------
if ( size(K) == 0 )
{
ring mr=0,x(1..n-1),dp;
ideal I=imap(nr,I);
H=maxdeg1(minbase(quotient(I,maxideal(1))))+1;
time=rtimer-time;
// Additional information:
dbprint(printlevel-voice+2,
"// The sequence of integers defines a monomial curve C in P"
+ string(n));
dbprint(printlevel-voice+2,
"// C is arithmetically Cohen-Macaulay");
dbprint(printlevel-voice+2,
"// Regularity attained at the last step of a m.g.f.r. of I(C)");
dbprint(printlevel-voice+2,
"// Regularity of the Hilbert function of S/I(C): "
+ string(H-2));
dbprint(printlevel-voice+2,
"// Time for computing ideal I(C) (by elimination): "
+ string(ttime) + " sec.");
dbprint(printlevel-voice+2,
"// Time for computing reg(C) once I(C) has been determined: "
+ string(time) + " sec.");
dbprint(printlevel-voice+2,
"// The Castelnuovo-Mumford regularity of C is:");
return(H);
}
else
{
KK=simplify(reduce(quotient(I,maxideal(1)),I),2);
firstind=maxdeg1(KK)+1;
J=subst(I,x(n),1);
ring mr=0,x(1..n-1),dp;
ideal J=imap(nr,J);
lastind=maxdeg1(minbase(quotient(J,maxideal(1))))+1;
H=firstind;
if ( lastind > H )
{
H=lastind;
}
time=rtimer-time;
// Additional information:
dbprint(printlevel-voice+2,
"// The sequence of integers defines a monomial curve C in P"
+ string(n));
dbprint(printlevel-voice+2,
"// C is not arithmetically Cohen-Macaulay");
dbprint(printlevel-voice+2,
"// end(H^1(S/I(C))) = "
+string(firstind-2));
dbprint(printlevel-voice+2,
"// Upper bound for the a-invariant of S/I(C): end(H^2(S/I(C))) <= "
+string(lastind-3));
if ( H == firstind )
{
dbprint(printlevel-voice+2,
"// Regularity attained at the last step of a m.g.f.r. of I(C)");
dbprint(printlevel-voice+2,
"// Regularity of the Hilbert function of S/I(C): "
+ string(H-1));
}
else
{
dbprint(printlevel-voice+2,
"// Regularity attained at the second last step of a m.g.f.r. of I(C)");
dbprint(printlevel-voice+2,
"// (and not attained at the last step)");
}
dbprint(printlevel-voice+2,
"// Time for computing ideal I(C) (by elimination): "
+ string(ttime) + " sec.");
dbprint(printlevel-voice+2,
"// Time for computing reg(C) once I(C) has been determined: "
+ string(time) + " sec.");
dbprint(printlevel-voice+2,
"// The Castelnuovo-Mumford regularity of C is:");
return(H);
}
}
example
{ "EXAMPLE:"; echo = 2;
// The 1st example is the twisted cubic:
regMonCurve(0,1,2,3);
// The 2nd. example is the non arithm. Cohen-Macaulay monomial curve in P4
// parametrized by: x(0)-s6,x(1)-s5t,x(2)-s3t3,x(3)-st5,x(4)-t6:
regMonCurve(0,1,3,5,6);
// Additional information is displayed if you change printlevel (=1);
}
////////////////////////////////////////////////////////////////////////////////
/*
Out-commented examples:
//
// The sequence of integers must be strictly increasing
regMonCurve(1,4,6,9);
regMonCurve(0,3,8,5,23);
regMonCurve(0,4,7,7,9);
//
// A curve in P3 s.t. the regularity is attained at the last step:
regMonCurve(0,2,12,15);
//
// A curve in P4 s.t. the regularity attained at the last but one
// but NOT at the last step (Ex. 3.3 Preprint 2004):
regMonCurve(0,5,9,11,20);
//
// A curve in P8 s.t. the m.g.f.r. of the defining ideal is not easily
// obtained through m.g.f.r.:
regMonCurve(0,1,2,3,9,11,18,24,25);
//
// A curve in P11 of degree 37:
regMonCurve(0,1,2,7,16,17,25,27,28,30,36,37);
// It takes some time to compute the eliminated ideal; the computation of
// the regularity is then rather fast as one can check using proc regIdeal:
ring q=0,(s,t,x(0..11)),dp;
ideal i=x(0)-st36,x(1)-s2t35,x(2)-s7t30,x(3)-s16t21,x(4)-s17t20,x(5)-s25t12,
x(6)-s27t10,x(7)-s28t9,x(8)-s30t7,x(9)-s36t,x(10)-s37,x(11)-t37;
ideal ei=eliminate(i,st);
ring qq=0,x(0..11),dp;
ideal i=imap(q,ei);
regIdeal(i);
//
// A curve in P14 of degree 55:
regMonCurve(0,1,2,7,16,17,25,27,28,30,36,37,40,53,55);
//
*/
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
proc NoetherPosition (ideal i)
"
USAGE: NoetherPosition (i); i ideal
RETURN: ideal such that, for the homogeneous linear transformation
map phi=S,NoetherPosition(i);
one has that K[x(n-d+1),...,x(n)] is a Noether normalization of
S/phi(i) where S=K[x(0),...x(n)] is the basering and d=dim(S/i).
(returns -1 if i = (0) or (1)).
ASSUME: The field K is infinite and i is a nonzero proper ideal.
NOTE: 1. It works also if K is a finite field if it terminates, but
may result in an infinite loop. If the procedure enters more
than 30 loops, -1 is returned and a warning message is displayed.@*
2. If printlevel > 0 (default = 0), additional info is displayed:
dim(S/i) and K[x(n-d+1),...,x(n)] are given.
EXAMPLE: example NoetherPosition; shows some examples
"
{
//--------------------------- initialisation ---------------------------------
int ii,jj,d,time,nl,NPtest;
intmat ran;
def r0 = basering;
ideal K,chcoord;
int n = nvars(r0)-1;
string s = "ring r1 = ",charstr(r0),",x(0..n),dp;";
execute(s);
ideal i,sbi,I,K,chcoord,m;
poly P;
map phi;
i = fetch(r0,i);
time=rtimer;
sbi=std(i);
I=simplify(lead(sbi),1);
attrib(I,"isSB",1);
d=dim(I);
//----- If the ideal i is not proper:
if ( d == -1 )
{
"// WARNING from proc NoetherPosition from lib mregular.lib:
// The ideal i is (1)!";
return (-1);
}
//----- If the ideal i is 0:
if ( size(I) == 0 )
{
"// WARNING from proc NoetherPosition from lib mregular.lib:
// The ideal i is (0)!";
return (-1);
}
//----- When the ideal i is 0-dimensional:
if ( d == 0 )
{
time=rtimer-time;
// Additional information:
dbprint(printlevel-voice+2,
"// Dimension of S/i: 0");
dbprint(printlevel-voice+2,
"// Time for computing a Noether normalization: "
+ string(time) + " sec.");
dbprint(printlevel-voice+2,
"// K is a Noether normalization of S/phi(i)");
dbprint(printlevel-voice+2,
"// where the map phi: S --> S is:");
setring r0;
return (maxideal(1));
}
NPtest=is_NP(I);
if ( NPtest == 1 )
{
K=x(n-d+1..n);
setring r0;
K=fetch(r1,K);
time=rtimer-time;
// Additional information:
dbprint(printlevel-voice+2,
"// Dimension of S/i: " + string(d) );
dbprint(printlevel-voice+2,
"// Time for computing a Noether normalization: " +
string(time) + " sec.");
dbprint(printlevel-voice+2,
"// K[" + string(K) +
"] is a Noether normalization of S/phi(i)");
dbprint(printlevel-voice+2,
"// where the map phi: S --> S is:");
return (maxideal(1));
}
//---- Otherwise, random change of coordinates and
//---- test for Noether normalization.
//---- If we were unlucky, another change of coord. will be done:
while ( nl < 30 )
{
chcoord=select1(maxideal(1),1..(n-d+1));
nl=nl+1;
for ( ii = 1; ii<=d; ii++ )
{
ran=random(100,1,n-d+ii);
ran=intmat(ran,1,n-d+ii+1);
ran[1,n-d+ii+1]=1;
m=select1(maxideal(1),1..(n-d+1+ii));
for ( jj = 1; jj<=n-d+ii+1; jj++ )
{
P=P+ran[1,jj]*m[jj];
}
chcoord[n-d+1+ii]=P;
P=0;
}
phi=r1,chcoord;
dbprint(printlevel-voice+2,"// (1 random change of coord.)");
I=simplify(lead(std(phi(i))),1);
attrib(I,"isSB",1);
NPtest=is_NP(I);
if ( NPtest == 1 )
{
K=x(n-d+1..n);
setring r0;
K=fetch(r1,K);
chcoord=fetch(r1,chcoord);
time=rtimer-time;
// Additional information:
dbprint(printlevel-voice+2,
"// Dimension of S/i: " + string(d) );
dbprint(printlevel-voice+2,
"// Time for computing a Noether normalization: " +
string(time) + " sec.");
dbprint(printlevel-voice+2,
"// K[" + string(K) +
"] is a Noether normalization of S/phi(i)");
dbprint(printlevel-voice+2,
"// where the map phi: S --> S is:");
return (chcoord);
}
}
"// WARNING from proc NoetherPosition from lib mregular.lib:
// The procedure has entered in more than 30 loops: in your example
// the method may enter an infinite loop over a finite field!";
return (-1);
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(x,y,z,t,u),dp;
ideal i1=y,z,t,u; ideal i2=x,z,t,u; ideal i3=x,y,t,u; ideal i4=x,y,z,u;
ideal i5=x,y,z,t; ideal i=intersect(i1,i2,i3,i4,i5);
map phi=r,NoetherPosition(i);
phi;
ring r5=5,(x,y,z,t,u),dp;
ideal i=imap(r,i);
map phi=r5,NoetherPosition(i);
phi;
// Additional information is displayed if you change printlevel (=1);
}
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
proc is_NP (ideal i)
"
USAGE: is_NP (i); i ideal
RETURN: 1 if K[x(n-d+1),...,x(n)] is a Noether normalization of
S/i where S=K[x(0),...x(n)] is the basering, and d=dim(S/i),
0 otherwise.
(returns -1 if i=(0) or i=(1)).
ASSUME: i is a nonzero proper homogeneous ideal.
NOTE: 1. If i is not homogeneous and is_NP(i)=1 then K[x(n-d+1),...,x(n)]
is a Noether normalization of S/i. The converse may be wrong if
the ideal is not homogeneous.
2. is_NP is used in the procedures regIdeal, depthIdeal, satiety,
and NoetherPosition.
EXAMPLE: example is_NP; shows some examples
"
{
//--------------------------- initialisation ---------------------------------
int ii,d,dz;
def r0 = basering;
int n = nvars(r0)-1;
string s = "ring r1 = ",charstr(r0),",x(0..n),dp;";
execute(s);
ideal i,sbi,I,J;
i = fetch(r0,i);
sbi=std(i);
I=simplify(lead(sbi),1);
attrib(I,"isSB",1);
d=dim(I);
//----- If the ideal i is not proper:
if ( d == -1 )
{
"// WARNING from proc is_NP from lib mregular.lib:
// The ideal i is (1)!";
return (-1);
}
//----- If the ideal i is 0:
if ( size(I) == 0 )
{
"// WARNING from proc is_NP from lib mregular.lib:
// The ideal i is (0)!";
return (-1);
}
//----- When the ideal i is 0-dimensional:
if ( d == 0 )
{
return (1);
}
//----- Check Noether position
J=I;
for ( ii = n-d+1; ii <= n; ii++ )
{
J=subst(J,x(ii),0);
}
attrib(J,"isSB",1);
dz=dim(J);
if ( dz == d )
{
return (1);
}
else
{
return(0);
}
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(x,y,z,t,u),dp;
ideal i1=y,z,t,u; ideal i2=x,z,t,u; ideal i3=x,y,t,u; ideal i4=x,y,z,u;
ideal i5=x,y,z,t; ideal i=intersect(i1,i2,i3,i4,i5);
is_NP(i);
ideal ch=x,y,z,t,x+y+z+t+u;
map phi=ch;
is_NP(phi(i));
}
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
proc is_nested (ideal i)
"
USAGE: is_nested (i); i monomial ideal
RETURN: 1 if i is of nested type, 0 otherwise.
(returns -1 if i=(0) or i=(1)).
ASSUME: i is a nonzero proper monomial ideal.
NOTES: 1. The ideal must be monomial, otherwise the result has no meaning
(so check this before using this procedure).@*
2. is_nested is used in procedures depthIdeal, regIdeal and satiety.@*
3. When i is a monomial ideal of nested type of S=K[x(0)..x(n)],
the a-invariant of S/i coincides with the upper bound obtained
using the procedure regIdeal with printlevel > 0.
THEORY: A monomial ideal is of nested type if its associated primes are all
of the form (x(0),...,x(i)) for some i<=n.
(see definition and effective criterion to check this property in
the preprint 'Saturation and Castelnuovo-Mumford regularity' by
Bermejo-Gimenez, 2004).
EXAMPLE: example is_nested; shows some examples
"
{
//--------------------------- initialisation ---------------------------------
int ii,d,tev,lastv,h,NPtest;
def r0 = basering;
int n = nvars(r0)-1;
string s = "ring r1 = ",charstr(r0),",x(0..n),dp;";
execute(s);
ideal I,K,KK,LL;
I = fetch(r0,i);
I=minbase(I);
attrib(I,"isSB",1);
d=dim(I);
//----- If the ideal i is not proper:
if ( d == -1 )
{
"// WARNING from proc is_nested from lib mregular.lib:
// The ideal i is (1)!";
return (-1);
}
//----- If the ideal i is 0:
if ( size(I) == 0 )
{
"// WARNING from proc is_nested from lib mregular.lib:
// The ideal i is (0)!";
return (-1);
}
//----- When the ideal i is 0-dimensional:
if ( d == 0 )
{
return (1);
}
//----- Check Noether position
NPtest=is_NP(I);
if ( NPtest != 1 )
{
return (0);
}
//----- When ideal is 1-dim. + var. in Noether position -> Nested Type
if ( d == 1 )
{
return (1);
}
//----- Determ. of the last variable really occuring
lastv=n-d;
h=n;
while ( lastv == n-d and h > n-d )
{
K=select(I,h+1);
if ( size(K) == 0 )
{
h=h-1;
}
else
{
lastv=h;
}
}
//----- Check the second property by evaluation when NP + d>1
KK=subst(I,x(lastv),1);
for ( ii = n-lastv; ii<=d-2; ii++ )
{
LL=minbase(subst(I,x(n-ii-1),1));
attrib(LL,"isSB",1);
tev=size(reduce(KK,LL));
if ( tev > 0 )
{
return(0);
}
KK=LL;
}
return(1);
}
example
{ "EXAMPLE:"; echo = 2;
ring s=0,(x,y,z,t),dp;
ideal i1=x2,y3; ideal i2=x3,y2,z2; ideal i3=x3,y2,t2;
ideal i=intersect(i1,i2,i3);
is_nested(i);
ideal ch=x,y,z,z+t;
map phi=ch;
ideal I=lead(std(phi(i)));
is_nested(I);
}
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
|