Opened 9 years ago

Closed 8 years ago

# Radical ideals in algebraic extensions

Reported by: Owned by: klooster@… pfister major 3-1-5 and higher dontKnow 3-1-5 radical

### Description

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);

I1=std(I1);
hilb(I1);

I2=std(I2);
hilb(I2);

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.
```

### comment:1 Changed 9 years ago by boehm

Owner: changed from somebody to pfister

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

### comment:2 Changed 8 years ago by hannes

Resolution: → fixed new → closed
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