Riv.Mat.Univ.Parma (7) 3* (2004)

AMEDEO SCREMIN

Exponential Diophantine equations and inequalities

Pages: 301-309
Mathematics Subject Classification (2000):
11D45 - 11D61

Abstract: Let us consider the ring of power sums with algebraic coefficients and positive integral roots, i.e. of functions of N of the form

(1)                                       a(n) = b1cn 1 + b2cn 2 + ... + bhcn h,

with biÎ Q and c1 > c2 >... > ch Î Z+.

Since long ago, Diophantine equations and inequalities involving power sums have been studied using the estimates for linear forms in logarithms due to A. Baker, but many problems remained unsolved. Recently P. Corvaja and U. Zannier have found new results on these problems by a different method, applying in this context the Subspace Theorem by W. Schmidt.
Here we will first have a review on some of such results, then we will show some recent results obtained by the author, partially with C. Fuchs. We will first deal with the finiteness of the solutions (n,y) Î NxZ of the equation
F(a(n),y) = f(n),
where FÎQ[X,Y] is monic, absolutely irreducible of degree at least 2, and f ÎZ[X] and the power sum a are not constant.
Then we will consider equations and inequalities where several power sums are involved as, for example, the equation

(2)                                       a0(n)yd + a1(n)yd-1 + ... + a d-1(n)y + ad(n) = 0

and the inequality

(3)                                         |F(n,y)|< a(n-d-1-e),

where a0, ..., ad are power sums, e > 0, F is monic in y and a is a quantity depending on the dominant roots of the power sums appearing as coefficients in F. We will show that, under suitable assumptions, for all the solutions of (2), y can be parametrized by some power sum in a finite set. We will reach a similar conclusion also for (3).

All these results generalize some results by P. Corvaja and U. Zannier on such problems.
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