A classification of restrictive polynomial correspondences

Abstract

In this manuscript, we study a special class of correspondences on ${\mathbb{P}}^1\times {\mathbb{P}}^1$ given by a polynomial relation, say $P(z,w)$. We focus on what we call restrictive polynomial correspondence and characterise that it can be written as $P(z,w)=g_1(w)h_1(z)+\cdots +g_{\ddot{I}}(w)h_{\ddot{I}}(z)$, for some appropriate $\ddot{I}\in {\mathbb{Z}}_{+}$, where $g_r$ and $h_r$ are polynomials. In particular, when $\ddot{I}=2$, we say $P$ is irreducible and observe that the equation $P(z,w)=0$ can be rewritten as $R(z)=S(w)$, where $R$ and $S$ are rational maps of appropriate degree. Further, we also define an operation that, with the exception of degenerate cases, constructs a new irreducible restrictive polynomial correspondence from any two given irreducible restrictive polynomial correspondences.

15 pages, Revisions done based on comments from the reviewer