The Skolem-Abouzaid theorem in the singular case

TL;DR

This paper extends Skolem's theorem on the finiteness of solutions to polynomial equations by removing the non-singularity condition at the origin, broadening its applicability to singular points.

Contribution

It generalizes Abouzaid's extension of Skolem's theorem by eliminating the non-singular point assumption at (0,0), allowing solutions near singular points on algebraic curves.

Findings

Bounded solutions in algebraic numbers are established.

Finiteness of solutions is proved for singular points.

The theorem's applicability is extended to more general cases.

Abstract

Let F(X;Y) in Q[X;Y] be a Q-irreducible polynomial. In 1929 Skolem proved the following theorem: "Assume that F(0;0) = 0. Then for every non-zero integer d, the equation F(X;Y) = 0 has only finitely many solutions in integers (X;Y) with gcd(X;Y) = d". Skolem method allows one to bound the solutions explicitly in terms of the coefficients of the polynomial F and the integer d. In 2008, Abouzaid gave a far-going generalization of Skolem theorem. He extended it in two directions: first, he studied solutions not only in rational integers, but in arbitrary algebraic numbers. Second, he not only bounded the solution in terms of the logarithmic gcd, but obtained a sort of asymptotic relation between the heights of the coordinates and their logarithmic gcd. Unfortunately, Abouzaid assumption is slightly more restrictive than Skolem: he assumes not only that the point (0;0) belongs to the plane curve F(X;Y) = 0, but that (0;0) is a non-singular point on this curve. The purpose of the present article is to get rid of this non singularity hypothesis.