ideal.groebner_basis gives incorrect answers

[broune]

This is wrong:

sage: R.<a,b,c,d>=PolynomialRing(QQ,order="lex")
sage: ideal(a-b^16,b-c^16,c-d^1024).groebner_basis()
[a - d^4096, b - d^16384, c - d^1024]

The correct answer as given by Macaulay 2:

i30 : R=QQ[a,b,c,d, MonomialOrder=>Lex];
i31 : I=ideal(a-b^16,b-c^16,c-d^1024);
i32 : gens gb I
o32 = | c-d1024 b-d16384 a-d262144 |

In particular the binomial involving a should raise d to the power 262144=2¹⁸, not 4096=2¹² as Sage reports.

I suspect that the reason is that by default Sage uses Singular to implement groebner_basis, and Singular has limitations on the size of exponents. See ​https://www.singular.uni-kl.de/Manual/4-0-2/sing_455.htm which in particular says

the maximal allowed exponent of a ring variable depends on the ordering of the ring
and is at least 32767.

(I suppose "least" is a typo, should be "most"...)

[jakobkroeker]

reported upstream:

​https://www.singular.uni-kl.de:8005/trac/ticket/774#ticket

[jakobkroeker]

another funny example :

R.<b,c,d>=PolynomialRing(QQ,order="lex")
sage: ideal(b-c^64,c-d^1024).groebner_basis()

(wrong result and zero not shown)

[b - , c - d^1024]

[dimpase]

Does Singular simply return incorrect results, or it's rather a Sage interface bug?

[jakobkroeker]

fixed in ​https://github.com/Singular/Sources/commit/349e5e898aad7e735b03bda328f2741a7c36a092

broken again for 'groebner' with ​https://github.com/Singular/Sources/commit/17510a828cdc33aa541d40a9bbfb825b72812521

[sbrandhorst]

sage: R.<a,b,c,d>=PolynomialRing(QQ,order="lex")
....: 
sage: I = ideal(a-b^16,b-c^16,c-d^1024)
sage: I.groebner_basis()
   skipping text from `)` error at token `)`
---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
RuntimeError: Error raised calling singular function
...
RuntimeError: error in Singular function call 'groebner':
exponent bound is 65535
error occurred in or before standard.lib::stdhilb line 381: `    i=interred(i);`
leaving standard.lib::stdhilb
leaving standard.lib::groebner

Looks good to me.

[dimpase]

This

  skipping text from `)` error at token `)`

looks ugly, but otherwise there seems nothing to fix here---unless nowadays Singular offers an option to have bigger exponents.