# multivariate polynomial ring over complex numbers

I want to factorize bivariate polynomials over C. For single variable case we do this as follow:

R=CC[x]
x=R.gen()
f=x^2+1
f.factor()


How to do this for multivariate case ?

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It seems not implemented on the floating-point complex numbers, nor on the complex algebraic numbers:

sage: R.<x,y> = CC[]
sage: P = x^2-y^2
sage: P.factor()
NotImplementedError: proof = True factorization not implemented.  Call factor with proof=False.
sage: P.factor(proof=False)
TypeError: Singular error:
? not implemented
? error occurred in or before STDIN line 26: def sage12=factorize(sage11);

sage: R.<x,y> = CDF[]
sage: P = x^2-y^2
sage: P.factor(proof=False)
NotImplementedError:

sage: R.<x,y> = QQbar[]
sage: P = x^2-y^2
sage: P.factor(proof=False)
TypeError: no conversion of this ring to a Singular ring defined


However, it is implemented for multivariate polynomials with integer or rational coefficients:

sage: R.<x,y> = ZZ[]
sage: P = x^2-y^2
sage: P.factor()
(x - y) * (x + y)


But note that the factorization is done with respect to the given ring, so you will get:

sage: P = x^2+1
sage: P.factor()
x^2 + 1


EDIT (July 2021)

Now, the factorization works on the field of algebraic numbers:

sage: R.<x,y> = QQbar[]
sage: P = x^2-y^2
sage: P.factor()
(x - y) * (x + y)

sage: P = x^2 + y^2
sage: P.factor()
(x + (-1*I)*y) * (x + 1*I*y)

sage: R.<x,y> = QQbar[]
sage: P = x^2+1
sage: P.factor()
(x - 1*I) * (x + 1*I)

sage: P = x^2+x*y+y^2
sage: P.factor()
(x + (0.50000000000000000? - 0.866025403784439?*I)*y) * (x + (0.50000000000000000? + 0.866025403784439?*I)*y)


This issue was likely fixed by trac ticket 25390

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Too bad that this is not possible. It would be nice to factorize x^2+y^2 or x^2+x*y+y^2 for example.