30 years of collaboration

Abstract

Abstract. We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem), which will be presented in turn.

Explicit bounds for composite lacunary polynomials

“…also triple-exponential) although it is quantitatively better than ours. Nevertheless, and also for the fact that our result is already mentioned in [2], it appears to us that the result is still worth to be found in the literature.…”

Explicit bounds for composite lacunary polynomials

“…Here, we deal with the situation that the number of terms of a given polynomial is fixed. It was conjectured by Erdős that if g is a complex non-constant polynomial with the property that g(x) 2 has at most l terms, then g(x) has also boundedly many terms and their number depends only on l. In [7], Schinzel proved a generalized version of Erdős's conjecture, namely the statement not only for g(x) 2 but for g(x) d , d ∈ N. He also extended the conjecture to compositions f (x) = g(h(x)), claiming that if f has l terms, then h(x) has at most B(l) terms for some function B on N. Zannier gave a proof for this (actually in a stronger version), wherein he showed in a first step the existence of a function B 1 such that, under the given assumptions, h(x) can be written as a rational function with at most B 1 (l) terms in both the numerator and the denominator [9]. Using this result, he proved the stated claim for representations as polynomials 1 and moreover, he gave a complete description of general decompositions f (x) = g(h(x)) of a given polynomial f (x) with l terms.…”