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resultant and gcd over real ?

asked 2014-02-12 11:03:49 -0600

updated 2014-02-12 13:16:21 -0600

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
a.resultant(b)
a.gcd(b)

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

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answered 2014-02-12 23:23:20 -0600

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

a.sylvester_matrix(b,x).det()

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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answered 2014-02-15 21:38:03 -0600

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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Asked: 2014-02-12 11:03:49 -0600

Seen: 130 times

Last updated: Feb 15 '14