The following theorem (lemma 4.2.18 on page 97) is proven in thesis “Computationally efficient Error-Correcting Codes and Holographic Proofs” by Daniel Alan Spielman:

Let $ E(X, Y)$ be a polynomial of degree ($ \alpha m, \beta n$ ) over some field $ \mathbb{F}$ and $ P(X, Y)$ be a polynomial of degree ($ \alpha m + \delta m, \beta n+\epsilon n$ ) over the same field. Assume there exist distinct $ x_1, \dots, x_m \in \mathbb{F}$ such that $ E(x_i, Y)$ divides $ P(x_i, Y)$ (as a polynomial in $ Y$ ) for $ 1 \leq i \leq m$ and distinct $ y_1, … y_n \in \mathbb{F}$ such that $ E(X, y_i)$ divides $ P(X, y_i)$ (as a polynomial in $ X$ ) for $ 1 \leq i \leq n$ .

If $ $ \alpha + \beta + \delta + \epsilon < 1\;\;\;\;\;(*)$ $ then $ E(X, Y)$ divides $ P(X,Y)$ in the ring $ \mathbb{F}[X, Y]$ .

My question is the following: how strict and precise is bound $ (*)$ ? Is there a counterexample to the statement, given that $ $ \alpha + \beta + \delta + \epsilon \geq 1?$ $

I suppose there are some connections with multivariate resultants and elimination theory, so I put them in the tags.