New York Journal of Mathematics
Volume 21 (2015) 987-1001


Dijana Kreso

On common values of lacunary polynomials at integer points

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Published: October 1, 2015
Keywords: Diophantine equation, lacunary polynomial, monodromy group, Morse polynomial, polynomial decomposition
Subject: 11D41, 12E05, 12F10

For fixed ℓ≧ 2, fixed positive integers m1> m2 with gcd(m1, m2)=1 and n1>n2> ... >n with gcd(n1, ..., n)=1, and fixed rationals a1, a2, ..., aℓ+1, b1, b2 which are all nonzero except for possibly aℓ+1, we show the finiteness of integral solutions x, y of the equation
when n1≧ 3, m1≧ 2ℓ(ℓ-1), and (n1, n2)≠ (m1, m2). In relation to that, we show the finiteness of integral solutions of equations of type f(x)=g(y), where f, g∈ Q[x] are of distinct degrees ≧ 3, and are such that they have distinct critical points and distinct critical values.


The author is thankful for the support of the Austrian Science Fund (FWF) via projects W1230-N13, FWF-P24302 and F5510.

Author information

Institut für Analysis und Computational Number Theory (Math A), Technische Universität Graz, Steyrergasse 30/II, 8010 Graz, Austria