 

Dijana Kreso
On common values of lacunary polynomials at integer points view print


Published: 
October 1, 2015

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


Abstract
For fixed ℓ≧ 2, fixed positive integers m_{1}> m_{2} with gcd(m_{1}, m_{2})=1 and n_{1}>n_{2}> ... >n_{ℓ} with gcd(n_{1}, ..., n_{ℓ})=1, and fixed rationals a_{1}, a_{2}, ..., a_{ℓ+1}, b_{1}, b_{2} which are all nonzero except for possibly a_{ℓ+1}, we show the finiteness of integral solutions x, y of the equation
a_{1}x^{n1}+...+a_{ℓ}x^{nℓ}+a_{ℓ+1}=b_{1}y^{m1}+b_{2}y^{m2},
when n_{1}≧ 3, m_{1}≧ 2ℓ(ℓ1), and (n_{1}, n_{2})≠ (m_{1}, m_{2}).
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.


Acknowledgements
The author is thankful for the support of the Austrian Science Fund (FWF) via projects W1230N13, FWFP24302 and F5510.


Author information
Institut für Analysis und Computational Number Theory (Math A), Technische Universität Graz, Steyrergasse 30/II, 8010 Graz, Austria
kreso@math.tugraz.at

