On a conjecture of Siegel

Abstract

Letk be an algebraically closed field of characteristic 0 and letf(x, y)∈k[t][x,y] be a polynomial in two variables with coefficients ink[t]. One is interested in solving the equationf(x,y)=0 with polynomialsx,y∈k[t]. Two solutions(x,y), (x′, y′) areproportional ifx′/x andy′/y are non-zero constants ink and a solution(x,y) isprimitive if the polynomialsx andy have no common root. The main result of this paper is that for a certain class of polynomialsf, which includes Thue equations with sufficiently lacunary exponents, the number of non-proportional, primitive solutions is bounded solely in terms of the number of monomials$a_i (t)x^{\alpha _1 } y^{\beta _i }$ appearing in the polynomialf(x,y). This verifies the analogue of a conjecture of Siegel for this class of polynomials. The proof is an application of theabc-theorem in function fields to certain determinantal varieties arising from the elimination of the coefficients of the polynomialf(x,y), together with an inductive argument on the numberr of monomials inf(x,y).