0

I'd like to know how to calculate the resultant of two bivariate polynomials. Searching through the web, I read that there is an Euclid-like algorithm for univariate polynomials, and that one can use it for bivariate polynomials by considering them as univariate polynomials with polynomial coefficients.

The only version of this Euclid-like algorithm I found is the one below (found on MSE), where LC means "leading coefficient":

F_1 := F
F_2 := G
i   := 2
R_1 := 1

while d_i:=deg(F_i) > 0:
   F_{i+1} := F_{i-1} mod F_i
   R_i := (-1)^{d_i * d_{i-1}} * LC(F_i)^{d_{i-1} - d_{i+1}} * R_{i-1}
   i++


if F_i != 0 then return R_{i-1} * LC(F_i)^{d_{i-1}}
   else return 0

However, in order to apply it to univariate polynomials with polynomial coefficients, I don't see how to get F_{i-1} mod F_i. Could you explain this point please? Or maybe orient me to another way, if this one is not the one to go?

Stéphane Laurent
  • 2,149
  • A link to the MSE post would be helpful. For now I can only guess that by mod they mean reduction $F_{i+1} = F_{i-1} - \frac{LT(F_{i-1})}{LT(F_{i})}F_{i}$, where $LT$ is the leading term with respect to a given monomial order (which is trivial in the univariate case). If you're looking for just a way to compute the resultant, taking the Sylvester determinant is probably the most straight-forward way. Otherwise there is literature on sophisticated fast algorithms for resultants of bivariate polynomials. – Randy Marsh Mar 15 '24 at 21:28
  • If this https://math.stackexchange.com/questions/696335/how-does-one-compute-a-resultant-by-using-euclidean-algorithm?rq=1 is the post you are referring to, note that the original poster says that the algorithm is incorrect – Randy Marsh Mar 15 '24 at 21:31
  • @RandyMarsh I fixed the mistake according to the answer. – Stéphane Laurent Mar 15 '24 at 21:33
  • I misunderstood the original algorithm. By mod they certainly mean Euclidean division. However, in the multivariate case mod is either replaced by reduction or subresultants. Reduction leads to Grobner basis, and there's vast literature for that. For a subresultant approach, see Chapter 8.3.6. of https://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted3.html (this is the academic website of one of the authors of the linked book). – Randy Marsh Mar 15 '24 at 22:01

0 Answers0