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.
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.