Thw following warning message looks pretty scary, since it is accompanied by a stack trace. And I have absolutely no idea (yet) as to what it is trying to tell me. Is the result it prints in the end reliable? If so, why the fuss? And if not, what is causing this problem? Is this an indication of a bug?
sage: R1.<cosAlpha, sinAlpha> = AA[]
sage: QR1 = R1.quotient(R1.ideal(cosAlpha^2 + sinAlpha^2 - 1))
sage: QR1(-8*sinAlpha - 4*sinAlpha*cosAlpha + 5*2*(2*sinAlpha*cosAlpha)*(cosAlpha*cosAlpha - sinAlpha*sinAlpha))
verbose 0 (3490: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.
singular_ring_delete(ring*) called with NULL pointer.
File "<stdin>", line 1, in <module>
File "_sage_input_4.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("UVIxKC04KnNpbkFscGhhIC0gNCpzaW5BbHBoYSpjb3NBbHBoYSArIDUqMiooMipzaW5BbHBoYSpjb3NBbHBoYSkqKGNvc0FscGhhKmNvc0FscGhhIC0gc2luQWxwaGEqc2luQWxwaGEpKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single')
File "", line 1, in <module>
File "/tmp/tmpKfaoEi/___code___.py", line 3, in <module>
exec compile(u'QR1(-_sage_const_8 *sinAlpha - _sage_const_4 *sinAlpha*cosAlpha + _sage_const_5 *_sage_const_2 *(_sage_const_2 *sinAlpha*cosAlpha)*(cosAlpha*cosAlpha - sinAlpha*sinAlpha))' + '\n', '', 'single')
File "", line 1, in <module>
File "sage/rings/quotient_ring.py", line 993, in _element_constructor_
return self.element_class(self, x)
File "sage/rings/quotient_ring_element.py", line 99, in __init__
self._reduce_()
File "sage/rings/quotient_ring_element.py", line 118, in _reduce_
self.__rep = I.reduce(self.__rep)
File "sage/rings/polynomial/multi_polynomial_ideal.py", line 4019, in reduce
strat = self._groebner_strategy()
File "sage/rings/polynomial/multi_polynomial_ideal.py", line 910, in _groebner_strategy
return GroebnerStrategy(MPolynomialIdeal(self.ring(), self.groebner_basis()))
Exception KeyError: (The ring pointer 0x0,) in 'sage.libs.singular.ring.singular_ring_delete' ignored
-40*cosAlphabar*sinAlphabar^3 + 16*cosAlphabar*sinAlphabar - 8*sinAlphabar
