This is a crosspost of <https://math.stackexchange.com/questions/446470/combinatorics-of-resultants> which received no answer. [**EDIT:** I deleted the initial copy of the question on MathSE].


Let $f(z)=\sum\_{i=0}^{D\_f}x\_iz^i$ and $g(z)=\sum\_{i=0}^{D\_g}y\_iz^i$ be two polynomials. I would like to know the number of monomials (in the variables $x\_i$ and $y\_i$) in the resultant in $z$ of $f$ and $g$. Equivalently, this is the number of monomials in the determinant of the Sylvester matrix:
$$
\left(
\begin{array}{ccccccccc}
x\_0 & x\_1 & x\_2 & \cdots & x\_{D\_f} & 0 & 0 & \cdots & 0\\
0 & x\_0 & x\_1 & \cdots & x\_{D\_f-1} & x\_{D\_f}& 0 & \cdots & 0 \\
\vdots & \ddots & \ddots &\cdots &\ddots &\ddots & \ddots &\cdots &\vdots\\
y\_0 & y\_1 & y\_2 & \cdots & y\_{D\_g} & 0 & 0 & \cdots & 0\\
0 & y\_0 & y\_1 & \cdots & y\_{D\_g-1} & y\_{D\_g}& 0 & \cdots & 0 \\
\vdots & \ddots & \ddots &\cdots &\ddots &\ddots & \ddots &\cdots &\vdots\\
\end{array}
\right).
$$


I think there is a classical answer to this problem (I even seem to recall having seen it once) but I can't find a pointer to it. Can anyone point me either to a closed form or even better a tight and simple upper bound on this number of monomials ?



Combinatorics of resultants
Yes, this (like everything else under the Sun) was studied, but not a lot. The reference is:
[M. Kalkbrener, An upper bound on the number of monomials in the Sylvester resultant.](https://www.evernote.com/shard/s24/sh/6c3eedc7-0e1e-483e-b5ca-baa54aa235a7/97920f99147371c0dc934d805c646704)


