Opened 6 years ago

## #21897 new defect

# Fix __div__ for multivariate polynomials in K[X][Y]

Reported by: | nborie | Owned by: | |
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Priority: | major | Milestone: | sage-7.5 |

Component: | commutative algebra | Keywords: | |

Cc: | Merged in: | ||

Authors: | Reviewers: | ||

Report Upstream: | N/A | Work issues: | |

Branch: | Commit: | ||

Dependencies: | Stopgaps: |

### Description

Polynomial Rings in any number of variables are able to symplify fraction in which the denominator divides the numerator. The following example shows e Newton divided difference (which contains a division but is always a polynomial, it works)

sage: K.<x1,x2,x3,x4> = PolynomialRing(QQ, 4) sage: P = x1^3*x2 - 8*x2*x3*x4^2; P x1^3*x2 - 8*x2*x3*x4^2 sage: Q = P.parent((P - P(x1,x3,x2,x4))/(x2-x3)); Q x1^3 sage: Q.parent() Multivariate Polynomial Ring in x1, x2, x3, x4 over Rational Field

Such symplification are not possible in K[X][Y] (it stay ok in K[X,Y]).

sage: K.<x> = PolynomialRing(QQ, 1) sage: L.<y> = PolynomialRing(K, 1) sage: y/y y/y sage: L(y/y) Traceback (most recent call last): TypeError: unable to coerce since the denominator is not 1 sage: y.divides(y) Traceback (most recent call last): TypeError: no conversion of this ring to a Singular ring defined sage: y/y == 1 True sage: y/y - 1 0 sage: (y/y).reduce() Traceback (most recent call last): ArithmeticError: unable to reduce because gcd algorithm doesn't work on input

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