# Linear Combination for Resultant

Let

R.<a, b, X> = ZZ[]
f = 1 - a*X^2
g = 1 - b*X^3


I need to compute polynomials u and v such that u f + v g = r where r = f.resultant(g, X).

Pari has a function polresultantext for that purpose, so one solution for my problem is

(u, v, r) = (R(p) for p in f._pari_().polresultantext(g._pari_(), 'X'))


Nevertheless, I have a few questions:

• Is there a method in Sage which directly does this?
• Is the detour via Pari the recommended solution? Or is there an alternative, e.g., in singular?
• If there is currently no method in Sage for doing this directly, what name would you recommend? resultant_ext ?
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• There is no such method yet, unfortunately.
• Singular does not seem to provide a method for this computation. See this page.
• There are discussions around this issue, for example on sage-devel and ticket #17674. Personally, I would opt for something around bezout_coefficients.
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