The Skolem-Abouzaïd theorem in the singular case

Abstract

Let $F(X,Y)\in \mathbb{Q}[X,Y]$ be a $\mathbb{Q}$-irreducible polynomial. In 1929, Skolem [13] proved a result allowing explicit bounding of the solutions of $F(X,Y)=0$ such that $\gcd (X,Y)=d$ in terms of the coefficients of $F$ and $d$. In 2008, Abouzaïd [1] generalized this result by working with arbitrary algebraic numbers and by obtaining an asymptotic relation between the heights of the coordinates and their logarithmic gcd. However, he imposed the condition that $(0,0)$ be a non-singular point of the plane curve $F(X,Y)=0$. In this paper, we remove this constraint.