inversion in polynomial quotient ring fails, but simply trying again makes it work

[yyyyx4]

Example:

F.<i> = GF((2**31-1,2), modulus=[1,0,1])
R.<x> = F[]
f = x^2 + (1196423630*i + 1564527877)*x + 2041534867*i + 2147483645
g = x^3 + x
S.<y> = R.quotient_ring(g)
print(~S(f))

Running this code in Sage 10.7 fails with the following error:

/usr/lib/python3.13/site-packages/sage/rings/polynomial/polynomial_quotient_ring_element.py:427: UserWarning: unexpected error from singular: ('characteristic is too large(max is 2^29)', 'zero divisor found - your minpoly is not irreducible', 'zero divisor found - your minpoly is not irreducible', '2nd module does not lie in the first')
  return type(self)(P, self._polynomial.inverse_mod(P.modulus()), check=False)
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
File sage/rings/polynomial/multi_polynomial.pyx:2042, in sage.rings.polynomial.multi_polynomial.MPolynomial.inverse_mod()

File sage/rings/polynomial/multi_polynomial_libsingular.pyx:4730, in sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular.lift()

ValueError: polynomial is not in the ideal

During handling of the above exception, another exception occurred:

ArithmeticError                           Traceback (most recent call last)
Cell In[1], line 6
      4 g = x**Integer(3) + x
      5 S = R.quotient_ring(g, names=('y',)); (y,) = S._first_ngens(1)
----> 6 print(~S(f))

File /usr/lib/python3.13/site-packages/sage/rings/polynomial/polynomial_quotient_ring_element.py:427, in PolynomialQuotientRingElement.__invert__(self)
    425 P = self.parent()
    426 try:
--> 427     return type(self)(P, self._polynomial.inverse_mod(P.modulus()), check=False)
    428 except ValueError as e:
    429     if e.args[0] == "Impossible inverse modulo":

File sage/rings/polynomial/polynomial_element.pyx:1658, in sage.rings.polynomial.polynomial_element.Polynomial.inverse_mod()

File sage/rings/polynomial/multi_polynomial.pyx:2045, in sage.rings.polynomial.multi_polynomial.MPolynomial.inverse_mod()

ArithmeticError: element is non-invertible

However, simply re-running the exact same command print(~S(f)) in the same Sage process seems to make it work after the first time. I suspect this is due to a problem with the Singular interface; see the errors/warnings in the first line of the output.

[yyyyx4]

(See §6.1 of the Singular manual for the upper bounds on the characteristic in different parts of the library.)

[user202729]

/usr/lib/python3.13/site-packages/sage/rings/polynomial/polynomial_quotient_ring_element.py:427: UserWarning: unexpected error from singular: ('characteristic is too large(max is 2^29)', 'zero divisor found - your minpoly is not irreducible', 'zero divisor found - your minpoly is not irreducible', '2nd module does not lie in the first')

looks like the fallback error checker is working as expected.

there are some existing ad hoc characteristic checks in the code base

[rings]$ rg '1 << 29'
polynomial/multi_polynomial_element.py
2191:            if base_ring.characteristic() > 1 << 29:
[rings]$ rg '1<<29'
polynomial/polynomial_element.pyx
5113:                    if R.characteristic() > 1<<29:

polynomial/multi_polynomial_libsingular.pyx
4213:        if n_GetChar(r.cf) > 1<<29:
5024:        if n_GetChar(_ring.cf) > 1<<29:
5131:        if n_GetChar(_ring.cf) > 1<<29:

one should probably refactor them into a common function. not to mention the ad hoc ones like

            finally:
                if check_error():
                    raise NotImplementedError("Factorization of multivariate polynomials over prime fields with characteristic > 2^29 is not implemented.")

that said, the blanket catch may not be that unreasonable, since factoring of monomial doesn't require factory library

sage: R.<x,y,z> = GF(2^32+15)[]
sage: (x*y).factor()
y * x

---

as far as I can tell if I use lift directly the same issue happens

F.<i> = GF((2**31-1,2), modulus=[1,0,1])
R.<x> = PolynomialRing(F, implementation="singular")
f = x^2 + (1196423630*i + 1564527877)*x + 2041534867*i + 2147483645
g = x^3 + x
u, v = R.one().lift((f, g))
f*u+g*v

so I'm tempted to conclude it's a singular bug.

[user202729]

I opened Singular/Singular#1292 .

For reference, one can modify interfaces/singular.py to add print(x) right before s = Expect.eval(self, x, **kwds), then do something like R.ideal((f, g))._singular_().lift(R.ideal((R.one()))._singular_()) in order to figure out what needs to be typed to the Singular interpreter to get the same behavior.