is_squarefree() over multivariate rings does not work

[sagetrac-MvanBeek]

The following examples demonstrate the problem:

sage: R.<x,y>=PolynomialRing(ZZ)
sage: R(1).is_squarefree()
False
sage: R(x^2+y^2).is_squarefree()
False
sage: R(x^2+1).is_squarefree()
False

CC: @malb @burcin @jpflori

Component: basic arithmetic

Keywords: squarefree, singular

Stopgaps: todo

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/12198

[vbraun]

comment:1

Singular seems to have issues with square-free factorization over ZZ:

                     SINGULAR                                 /  Development
 A Computer Algebra System for Polynomial Computations       /   version 3-1-3
                                                           0<
 by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann     \   March 2011
FB Mathematik der Universitaet, D-67653 Kaiserslautern        \
> ring R = integer,(x,y), dp;
> poly p = x^2+y^2;
> sqrfree(p);
   ? not implemented
   ? error occurred in or before STDIN line 3: `sqrfree(p);`

This might be implemented in a more recent version, don't know about that. Looking at the Singular source there is a isSqrFreeZ function in the factory that should do the right thing. Its not clear to me why the code path doesn't end up there.

Over QQ it works fine:

> ring Q = 0,(x,y),dp;
> poly p = x^2+y^2;
> sqrfree(p);
_[1]=x2+y2

I haven't found a way to test square-freeness (as opposed to compute the squarefree factorization) from the Singular command line, is this function not exposed to the Singular interpreter?

[burcin]

comment:3

Integer coefficients are relatively new in Singular. Many fuctions do not work with them. In this case, Singular raises an error using a callback function (see sage.libs.singular.singular.libsingular_error_callback). Unfortunately we do not check for it and return wrong results.

A workaround, suggested by Martin Lee, for this immediate problem is to change_ring(QQ) and call is_squarefree() there.

If there isn't a ticket already about handling errors raised in libsingular, we should open one. Note that the magical function interface (the one you get through sage.libs.singular.ff.*) can handle Singular errors. It's only direct C library calls that should check explicitly if Singular raised an error. Perhaps we can solve it by wrapping the declarations in sage/libs/singular/singular-cdefs.pxi in a preprocessor macro which calls PyErr_Occured() after raising an exception in libsingular_error_callback().

[burcin]

Changed keywords from none to squarefree, singular

[zimmermann6]

comment:4

the same problem happens over finite fields:

sage: R.<x,y> = GF(2)[]
sage: f=x+y
sage: f.is_squarefree()
False

If is_squarefree is wrong, it should be simply disabled.

See also #12404.

Paul

[zimmermann6]

comment:5

see also #12129.

Paul