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 ?

heightof the polynomial, is discussed in Carlos D'Andrea and Kevin G. Hare, On the height of the Sylvester resultant, Experimental Mathematics, 13:3 (2004) 331-341. $\endgroup$ – Gerry Myerson Jul 23 '13 at 1:11