# Division of bivariate polynomials

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.