I am trying to write a program that computes the resultant of two polynomials with coefficients in $\mathbb{Z}[Y]$. If I know the degree of the polynomial as a polynomial in $Y$ is $k$ say, I can evaluate at $k+1$ points by computing resultants over $\mathbb{Z}$ which I can already do, but how do I know what the degree of the resultant over $\mathbb{Z}[Y]$ is?
I can bound it by the product of the $Y$-degrees of the input polynomials. It seems to me like this bound shouldn't always be achieved.
Is there a way around this, either by not needing to know the exact degree, or by finding what the degree is easily?