Let $f,g \in F[x,y], \deg_xf=n,\deg_xg=m,d\in \mathbb{N}:\deg_y f, \deg_y g \leq d$. Then $\deg_y res_x(f,g) \leq (n+m)d$.
I have $fg=\sum_{i=0}^n \sum_{j=0}^m f_{ij} g_{ij} x^i y^j$ (Is this a correct form?) Now for the resultant I somehow need to extract the coefficients that belong to $x$? I'm struggling to show the inequality with the degrees.
Thanks for any help!
If $f$ has degree $n$ and $g$ has degree $m$, the resultant $res(f, g)$ of $f$ and $g$ is the determinant of the $(m+n) \times (m+n)$ matrix whose first column is the coefficients of $f$ padded by $0$ until the end of the column, the next is this one shifted down by one with a zero on top and so on until the leading coefficient of $f$ "touches the bottom of the matrix" and then we proceed the same with the coefficients of $g$. Let's denote this matrix by $(a_i^j)_{1 \leq i, j \leq m+n}.$ In our case, we are interested in the resultant of the two polynomials in the variable $x$. Observe that each entry in this determinant is then a polynomial in $y$ of degree at most $\max (\deg_y f, \deg_y g) \leq d$ by hypothesis. Then by the definition of the determinant, $res_x (f, g) = \det((a_i^j)_{1 \leq i, j \leq m+n}) = \sum_{\sigma \in S_{m+n}} (-1)^{sign(\sigma)} \prod_{1 \leq i \leq m+n} a_i^{\sigma(i)}.$ It follows that $\deg_y \prod_{1 \leq i \leq m+n} a_i^{\sigma(i)} \leq (m + n)d,$ which in turn implies that $\deg_y res_x (f, g) \leq (m + n) d.$ Hope this helps. :)