Radical ideals in algebraic extensions

[klooster@…]

There are at least three different ways to compute the radical of an ideal in Singular. In the example below, for each way to compute the radical I obtain an ideal with a different Hilbert function. This is probably due to the fact I work in an algebraic extension. However, the manual does not exclude/warn that radical or radicalEHV does not work over algebraic extensions of the rationals. The geometry behind the example suggests that radicalEHV gives the correct output.

LIB "primdec.lib";
ring R=(0,a),(x,y,z),dp;
minpoly=a2+a+1;

poly L=x;
poly u1=1;
poly u2=1;
poly v1=y;
poly v2=z;
poly g1a=(a+2)/(-3)*(u1*v1*a-u2*v2);
poly g2a=(a+2)/(-3)*(u1*v1-u2*v2);
poly g1=L*g1a;
poly g2=L*g2a;
poly v3=g1a+(a+1)*g2a;
poly f1=(u1*u2*L^3+v1*v2*v3)/2;
poly f2=(-u1*u2*L^3+v1*v2*v3)/2;

poly F=g1^3-f1^2;
ideal I=jacob(F);
I=std(I);
hilb(I);

ideal I1=radical(I,"KL");
I1=std(I1);
hilb(I1);

ideal I2=radical(I);
I2=std(I2);
hilb(I2);


ideal I3=radicalEHV(I);
I3=std(I3);
hilb(I3);
quit;
---
                     SINGULAR                                 /
 A Computer Algebra System for Polynomial Computations       /   version 3-1-5
                                                           0<
 by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann     \   Jul 2012
FB Mathematik der Universitaet, D-67653 Kaiserslautern        \
// ** loaded /usr/share/Singular/LIB/primdec.lib (14732,2012-03-30)
// ** loaded /usr/share/Singular/LIB/ring.lib (15100,2012-07-10)
// ** loaded /usr/share/Singular/LIB/absfact.lib (14191,2011-05-04)
// ** loaded /usr/share/Singular/LIB/triang.lib (13499,2010-10-15)
// ** loaded /usr/share/Singular/LIB/matrix.lib (13658,2010-11-16)
// ** loaded /usr/share/Singular/LIB/nctools.lib (14246,2011-05-26)
// ** loaded /usr/share/Singular/LIB/inout.lib (13499,2010-10-15)
// ** loaded /usr/share/Singular/LIB/random.lib (14661,2012-03-05)
// ** loaded /usr/share/Singular/LIB/poly.lib (14852,2012-04-30)
// ** loaded /usr/share/Singular/LIB/elim.lib (14661,2012-03-05)
// ** loaded /usr/share/Singular/LIB/general.lib (14191,2011-05-04)
//         1 t^0
//        -3 t^5
//         3 t^8
//        -1 t^9

//         1 t^0
//         2 t^1
//         3 t^2
//         4 t^3
//         5 t^4
//         3 t^5
//         1 t^6
//        -1 t^7
// dimension (proj.)  = 0
// degree (proj.)   = 18
//         1 t^0
//        -1 t^2
//        -2 t^3
//         2 t^4

//         1 t^0
//         2 t^1
//         2 t^2
// dimension (proj.)  = 0
// degree (proj.)   = 5
//         1 t^0
//        -3 t^4
//         2 t^6

//         1 t^0
//         2 t^1
//         3 t^2
//         4 t^3
//         2 t^4
// dimension (proj.)  = 0
// degree (proj.)   = 12
//         1 t^0
//        -1 t^3
//        -3 t^4
//         3 t^5

//         1 t^0
//         2 t^1
//         3 t^2
//         3 t^3
// dimension (proj.)  = 0
// degree (proj.)   = 9
Auf Wiedersehen.

[boehm]

Hallo Gerhard, siehst Du, was hier das Problem ist?

[hannes]

fixed by Pfister with 3a823d31cc10c360f47890e903f0104a00e4f97b