polynomial rings should not attempt to use Singular in characteristic >= 2^31

[yyyyx4]

Example:

sage: R.<x,y> = GF(2147483659^2)[]
sage: x.resultant(y)
# [...]
TypeError: Singular error:
   ? Wrong or unknown ground field specification
   ? cannot make ring
   ? error occurred in or before STDIN line 14: `ring sage8=(2147483659,z2),(x, y),dp;`
   ? expected ring-expression. type 'help ring;'

Reason: can_convert_to_singular() only checks that the characteristic is small for prime fields, but Singular equally can't handle finite extension fields of large characteristic.

On the other hand, Singular can handle arbitrarily large IntegerModRings (including prime finite fields) using the (integer,modulus) syntax.

Component: algebra

Author: Lorenz Panny

Branch/Commit: eafaba1

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/33319

[yyyyx4]

Description changed:

--- 
+++ 
@@ -11,5 +11,7 @@
    ? expected ring-expression. type 'help ring;'
 ```
 
-Reason: `can_convert_to_singular()` only checks that the characteristic is small for *prime* fields, but Singular equally can't handle base rings of large composite characteristic and non-prime finite fields of large characteristic.
+Reason: `can_convert_to_singular()` only checks that the characteristic is small for *prime* fields, but Singular equally can't handle finite extension fields of large characteristic.
 
+On the other hand, Singular *can* handle arbitrarily large `IntegerModRing`s (including prime finite fields) using the `(integer,modulus)` syntax.
+

[yyyyx4]

comment:2

The individual commits should be easier to understand than the overall diff.

---

New commits:

b5f068e	deduplicate `_singular_init_` code
16f05a8	deduplicate many close-to-identical singular.ring() calls
60f42c5	discontinue calling singular.ring() with an integer argument; this also makes check= obsolete
1db734e	Singular supports characteristic >= 2^31 for ZZ/m but not for non-prime finite fields
d51787c	some code cleanup (not functional changes)

[yyyyx4]

Commit: d51787c

[yyyyx4]

Branch: public/fix_singular_conversions_int32_limit

[yyyyx4]

Author: Lorenz Panny

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

b5417f0	deduplicate `_singular_init_` code
e57f750	deduplicate many close-to-identical singular.ring() calls
c5ec68c	discontinue calling singular.ring() with an integer argument; this also makes check= obsolete
440ced3	Singular supports characteristic >= 2^31 for ZZ/m but not for non-prime finite fields
ad9661b	some code cleanup (no functional changes)

[sagetrac-git]

Changed commit from d51787c to ad9661b

[sagetrac-git]

Changed commit from ad9661b to c490d80

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

c490d80

these doctests now use Singular, hence no more "toy implementation" warning

[tscrim]

comment:5

I am not sure I entirely like moving something from Cython that could be called a lot to a Python function. I understand it would be easier to maintain, but I am not sure it is worth the potential speed tradeoff since this is a fairly important part of Sage. This isn't a strong opinion, but it is something to consider.

There are a number of doctest failures. Most appear to be trivial, but they might be subtly indicating something. The adding of the fallback warnings might need some justification.