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resultant and gcd over real ?

asked 2014-02-12 18:03:49 +0200

etienne gravatar image

updated 2014-02-12 20:16:21 +0200

slelievre gravatar image

The resultant and gcd work well over rational but not over real or complex!

R.<x,y> = PolynomialRing(RR,2)
a = x^2 + y
b = x - y^2

This does not work while with QQ, it works. Anyone has an idea?

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answered 2014-02-16 04:38:03 +0200

Jeroen Demeyer gravatar image

If you define your polynomial ring as follows, it does work:

sage: P.<y> = PolynomialRing(RR)
sage: R.<x> = PolynomialRing(P)
sage: a = x^2 + y
sage: b = x - y^2
sage: a.resultant(b)
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answered 2014-02-13 06:23:20 +0200

lftabera gravatar image

The problem is that for these kind of inexact rings the methods are not implemented.

For the resultant, you can compute the resultant with


But I cannot promise performance nor numeric stability.

The gcd is more tricky. For instance, if you take univariate polynomials over an inexact rings and perform Euclides method, you will end with a gcd 1 due to numerical errors. You are asking for an approximate gcd which is not easy to compute.

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Asked: 2014-02-12 18:03:49 +0200

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Last updated: Feb 16 '14