Polynomial bounds for the solutions of a class of diophantine equations
Let \(K\) be an algebraic number field of degree \(d\) such that all embeddings of \(K\) into \(\mathbb{C}\) are real. Let \(F(X,Y)\in K[X,Y]-K[Y]\) be an irreducible polynomial of total degree \(N\) and of degree \(n>0\) in \(Y\). Denote by \(F_N(X,Y)\) the homogeneous part of degree \(N\) of \(F(X,Y)\) and assume that \(F_N(1,Y)\) is of degree \(n\) without real roots. Consider the equation \[ F(x,y)=0 \text{ in} (x,y)\in O_K\times K. \] In the main result of the paper the author proves that the solutions of this equation satisfy \[ \max(H_K(x),H_K(y))<(2^N N)^{4dN^3} H_K(F)^{2N^3}. \] This is maybe the first polynomial bound for the height of integer points of a class of algebraic curves over a number field. In the proof, Puiseux expansions of algebraic functions are used.
polynomial bounds for the height of integer points
scientific article; zbMATH DE number 1088553
scientific article