lots of PolynomialError contains an element of the set of generators exceptions calling integrate in sympy 1.12

[nasser1]

I do not know if this is known or reported before. Feel free to close if so.

I see lots of exceptions from polyutils.py calling integrate command. Below is just one typical example

from sympy import *
x=symbols('x')
integrand="(-36-2*6**(1/2)*x**2)/(9*3**(1/2)+3*2**(1/2)*x**2-x**4)"
integrate( sympify(integrand), x)

gives

Traceback (most recent call last):
  File "/usr/lib/python3.11/site-packages/sympy/polys/polyutils.py", line 211, in _parallel_dict_from_expr_if_gens
    monom[indices[base]] = exp
          ~~~~~~~^^^^^^
KeyError: 1/(2*_t**4 - 2*sqrt(2)*_t**2 + 1)

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "/usr/lib/python3.11/site-packages/sympy/integrals/integrals.py", line 1567, in integrate
    return integral.doit(**doit_flags)
           ^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/usr/lib/python3.11/site-packages/sympy/integrals/integrals.py", line 612, in doit
    antideriv = self._eval_integral(
                ^^^^^^^^^^^^^^^^^^^^
  File "/usr/lib/python3.11/site-packages/sympy/integrals/integrals.py", line 955, in _eval_integral
    result, i = risch_integrate(f, x, separate_integral=True,
                ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/usr/lib/python3.11/site-packages/sympy/integrals/risch.py", line 1835, in risch_integrate
    ans = integrate(fa.as_expr()/fd.as_expr(), DE.x, risch=False)
          ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/usr/lib/python3.11/site-packages/sympy/integrals/integrals.py", line 1567, in integrate
    return integral.doit(**doit_flags)
           ^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/usr/lib/python3.11/site-packages/sympy/integrals/integrals.py", line 612, in doit
    antideriv = self._eval_integral(
                ^^^^^^^^^^^^^^^^^^^^
  File "/usr/lib/python3.11/site-packages/sympy/integrals/integrals.py", line 1041, in _eval_integral
    parts.append(coeff * ratint(g, x))
                         ^^^^^^^^^^^^
  File "/usr/lib/python3.11/site-packages/sympy/integrals/rationaltools.py", line 83, in ratint
    L = ratint_logpart(r, Q, x, t)
        ^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/usr/lib/python3.11/site-packages/sympy/integrals/rationaltools.py", line 231, in ratint_logpart
    res = Poly(res, t, composite=False)
          ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/usr/lib/python3.11/site-packages/sympy/polys/polytools.py", line 182, in __new__
    return cls._from_expr(rep, opt)
           ^^^^^^^^^^^^^^^^^^^^^^^^
  File "/usr/lib/python3.11/site-packages/sympy/polys/polytools.py", line 311, in _from_expr
    rep, opt = _dict_from_expr(rep, opt)
               ^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/usr/lib/python3.11/site-packages/sympy/polys/polyutils.py", line 368, in _dict_from_expr
    rep, gens = _dict_from_expr_if_gens(expr, opt)
                ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/usr/lib/python3.11/site-packages/sympy/polys/polyutils.py", line 307, in _dict_from_expr_if_gens
    (poly,), gens = _parallel_dict_from_expr_if_gens((expr,), opt)
                    ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/usr/lib/python3.11/site-packages/sympy/polys/polyutils.py", line 216, in _parallel_dict_from_expr_if_gens
    raise PolynomialError("%s contains an element of "
sympy.polys.polyerrors.PolynomialError: 1/(2*_t**4 - 2*sqrt(2)*_t**2 + 1) contains an element of the set of generators.
>>> 

Here is another example

from sympy import *
x=symbols('x')
integrand="(3-2*2**(1/2)+x**2)**2*(-3+2*2**(1/2)+x**2)/(577-408*2**(1/2)-8* (-41+29*2**(1/2))*x**2-2*(-39+28*2**(1/2))*x**4-8*(2**(1/2)-1)*x**6+x**8)"
integrate( sympify(integrand), x)

gives

 File "/usr/lib/python3.11/site-packages/sympy/polys/polyutils.py", line 216, in _parallel_dict_from_expr_if_gens
    raise PolynomialError("%s contains an element of "
sympy.polys.polyerrors.PolynomialError: 1/(-4893319121142556020618924174780472498117708482611714912381696*_t**4 + 3460099133069698398004476359279702930052248019321310378430976*sqrt(2)*_t**4 - 159769239484575670917838951113184628965915778476312975023968*_t**2 + 112973912664560957914881642683593608324699445462537847255936*sqrt(2)*_t**2 - 1304135764168161431882534016166025912200366086490350062499 + 922163242431207071074983301530248435746125497723607601750*sqrt(2)) contains an element of the set of generators.

And another example

from sympy import *
x=symbols('x')
integrand="(-3+2*2**(1/2)+x**2)/(17-12*2**(1/2)+(2-4*2**(1/2))*x**2+x**4)"
integrate( sympify(integrand), x)

Gives

 File "/usr/lib/python3.11/site-packages/sympy/polys/polyutils.py", line 216, in _parallel_dict_from_expr_if_gens
    raise PolynomialError("%s contains an element of "
sympy.polys.polyerrors.PolynomialError: 1/(-2304*_t**4 + 1024*sqrt(2)*_t**4 - 32*_t**2 + 64*sqrt(2)*_t**2 - 1) contains an element of the set of generators.

Version info

>>> import sympy
>>> sympy.__version__
'1.12'

>python --version
Python 3.11.8
>

[oscarbenjamin]

Somehow res here is not a polynomial in t:

sympy/sympy/integrals/rationaltools.py

Lines 225 to 231 in d11d0b2

	f, g = Poly(f, x), Poly(g, x)
	t = t or Dummy('t')
	a, b = g, f - g.diff()*Poly(t, x)
	res, R = resultant(a, b, includePRS=True)
	res = Poly(res, t, composite=False)

Something very string happening here:

ipdb> p a
Poly(x**4 - 3*sqrt(2)*x**2 - 9*sqrt(3), x, domain='EX')
ipdb> p b
Poly(-4*_t*x**3 + 2*sqrt(6)*x**2 + 6*sqrt(2)*_t*x + 36, x, domain='EX')
ipdb> p resultant(a, b)
-1213056*sqrt(3)*_t**8/(2*_t**4 - 2*sqrt(2)*_t**2 + 1) - 1119744*_t**8/(2*_t**4 - 2*sqrt(2)*_t**2 + 1) + 2239488*sqrt(2)*_t**6/(2*_t**4 - 2*sqrt(2)*_t**2 + 1) + 2426112*sqrt(6)*_t**6/(2*_t**4 - 2*sqrt(2)*_t**2 + 1) - 1912896*sqrt(3)*_t**4/(2*_t**4 - 2*sqrt(2)*_t**2 + 1) + 839808*_t**4/(2*_t**4 - 2*sqrt(2)*_t**2 + 1) - 3079296*sqrt(2)*_t**2/(2*_t**4 - 2*sqrt(2)*_t**2 + 1) - 513216*sqrt(6)*_t**2/(2*_t**4 - 2*sqrt(2)*_t**2 + 1) + 559872*sqrt(3)/(2*_t**4 - 2*sqrt(2)*_t**2 + 1) + 1819584/(2*_t**4 - 2*sqrt(2)*_t**2 + 1)

The resultant should not have denominators like this. Maybe resultant is broken in the EX domain. Probably because EX uses cancel and cancel cannot handle sqrt(2) and sqrt(3) without extension=True. For EX probably a division free resultant algorithm should be used like actually computing the determinant of the Sylvester matrix or something.