ISSN:
1436-5081
Keywords:
11D99
;
Diophantine equations
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01305344
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