Find gcd of two multivariate polynomials

Question

Is there a simple way to find the $\gcd$ of $x^2y$ and $xy^2+1$?

I tried adding multiples of $x^2y$ to the other and vice-versa but I found no easy way to find the gcd. For another example I found $(xy,x^3y+1)=(xy,x^3y+1-x^3y)=1$ by adding $-x^2$ of the first to the second.

Well, it would have to be a divisor of $x^2y$ and there aren't very many of those. — lulu, Mar 18 '24 at 16:42
Since $x$ and $y$ are coprime to $xy^2+1$ so too is the product $x^2 y$, by here in the linked dupe. $\ \ $ — Bill Dubuque, Mar 18 '24 at 18:36

score 3 · Answer 1 · edited Mar 18 '24 at 16:54

3

You can use Buchberger's algorithm to find such a polynomial if it exists. The algorithm is suited to find sets of generating multivariate polynomials for a given ideal, the so called Gröbner bases.

edited Mar 18 '24 at 16:54

Sammy Black

28,409

answered Mar 18 '24 at 16:52

Marius S.L.

2,501

Not an answer, and serious overkill for such a simple problem. – Bill Dubuque Mar 18 '24 at 18:37

Find gcd of two multivariate polynomials

1 Answers1