  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.
