zero coefficients after differentiation in positive characteristic

[Forum]

Hello all,

this seems to be a bug in the second variant of diff. Consider:

ring r=2,(x),dp; print(diff(x2,x)); -> 0 print(diff(ideal(x),ideal(x2))); BUG: 0x print(diff(ideal(x),ideal(x2+x,x4+x))); BUG: 0x+1,0x3+1

It seems to forget to check the resulting coefficient to be zero. Must be easy to fix by looking at the 1st variant.

Sincerely Yours,

Oleksandr Motsak

On Tue, Jan 12, 2010 at 8:37 PM, <owner-singular-team@…> wrote:

Gruß

Alexander

​http://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=1794&sid=c8b4347d4c18c087e097a2a6803fc177 kedlaya

Post subject: Differentiation in positive characteristic

Posted: Tue Jan 12, 2010 6:42 pm

I believe I have found a bug in the command "diff", when working in positive characteristic.

The command "diff" can be used in two different ways. One is to differentiate with respect to a single variable at a time, and this seems to work fine also in positive characteristic.

ring r=0,(x,y),dp; print(diff(x2,x));

2x

ring r=2,(x,y),dp; print(diff(x2,x));

0

The other is to differentiate several polynomials with respect to several variables, and this seems to have trouble in positive characteristic. (Note that the syntax is reversed in this usage, with the variables coming first. I find that odd, but it's not the issue at hand.)

ring r=0,(x,y),dp; ideal H = x,y; ideal I = x2,y2; print(diff(H,I));

2x,0, 0, 2y

ring r=2,(x,y),dp; ideal H = x,y; ideal I = x2,y2; print(diff(H,I));

0x,0, 0, 0y

The trouble is that the diagonal entries are supposed to be zero. Their coefficients have been reduced modulo 2, but the resulting zeroes have not been removed. In fact, they are not recognized as being equal to zero!

print(diff(H,I)[1,1] != 0);

1 -- Dr. rer. nat. Dipl.-Math. Alexander Dreyer

Abteilung "Systemanalyse, Prognose und Regelung" Fraunhofer Institut für Techno- und Wirtschaftsmathematik (ITWM) Fraunhofer-Platz 1 67663 Kaiserslautern

Telefon +49 (0) 631-31600-4318 Fax +49 (0) 631-31600-1099 E-Mail alexander.dreyer@… Internet ​http://www.itwm.fhg.de/de/as/indexas/

[hannes]

fixed, test file added