## Setup

In his thesis (lemma 4.2.18, p. 97-98) Spielman describes a divisibility test for bivariate polynomials $E,P\in k[X,Y]$, where $k$ is a field (of positive characteristic for what I'm interested in). It is a converse to the observation that if $E\mid P$ in $k[X,Y]$, then for any $u\in k$, substituting $X$ with $u$ yields a divisibility relation $E(u,Y)\mid P(u,Y)$ within $k[Y]$. So as to not distract from my actual question, I state a vague version of his result.

Lemma.If there are enough $x_1,\dots,x_m\in k$ and $y_1,\dots,y_n\in k$ such that $E(x_i,Y)\mid P(x_i,Y)$ and $E(X,y_j)\mid P(X,y_j)$ then $E\mid P$.

The proof works in two steps: prove the assertion for coprime polynomials $E$ and $P$, and reduce the general case to the coprime case. In the coprime case, he considers $E,P\in R[Y]$ to be polynomials with coefficients in the ring $R=k[X]$: $$ E=\sum_{i=0}^{\deg_Y(E)}E_i(X) Y^i,\quad P=\sum_{i=0}^{\deg_Y(P)}P_i(X) Y^i $$ he associated resultant ($\deg_Y(E)=\beta n$ and $\deg_Y(P)=(\beta+\epsilon)n$ in his notation) $\mathrm{Res}(E,P)=k[X]$ must be nonzero and there is an easy bound on the degree. The gist of the proof is to show that $R$ has too many roots counted with multiplicities to be nonzero, and to derive a contradiction from there.

## My problem

It seems the justification for the multiplicity count given in the paragraph below is lacking in two ways

- I see no reason why there should be the claimed linear dependence. What is preventing a situation where say $E(x_i,Y)=1$ and $\deg_Y(P(x_i) = \deg_Y(P)$ for instance (in which case the matrix $M(E,P)(x_i)$ is invertible and $x_i$ is not a root of $\mathrm{Res}(E,P)$) ? All I can safely say is that if $\deg_Y P - \deg_Y P(x_i,Y) \geq \deg_Y E - \deg_Y E(x_i,Y)$ then the claim is true.
- There is another difficulty arising from the fact that Spielman uses vanishing of derivatives to obtain information about multiplicities, but it was suggested to me that one can salvage the argument using ``hyper derivatives'' and this seems plausible.

## Questions

- Are these concerns valid?
- Is there a way to resolve the first one?
- Are there missing assumptions?
- Has this result been written up and corrected elsewhere?

Also please feel free to migrate this question to the theoretical computer science stack.exchange if necessary. There already is a question about that proposition on there, but the question is different.