PhD Thesis

Finiteness results for Diophantine equations for polynomial families  (Stoll Thomas) (ps)  
We study the general Diophantine equation A P_m(x)+ B P_n(y)=C in integers x, y, where A, B, C are fixed rational numbers and P_m(x) and P_n(y) are polynomials of fixed degrees m and n taken from a given polynomial family {P_k(x)}. There are various connections with problems that have so far not been entirely solved. Mention, for instance, the classical question, whether there are finitely or infinitely many integers x, y such that their falling products of length m and n respectively, are equal. We use an algorithmic criterion of Bilu and Tichy in order to decide finiteness (in terms of number of solutions (x,y)) as well as for other polynomial families, e.g. general Meixner and Krawtchouk polynomials, polynomials with zeroes in arithmetic progressions, polynomials which satisfy three-term recurrences etc. In particular, we obtain finiteness for all known continuous classical orthogonal polynomials except for the Chebyshev polynomials. It is well-known that the latter polynomials play a very special role in various fields of mathematics (i.e. approximation theory, numerical analysis). Interestingly, they turn out to be exceptional in this Diophantine context, too.
 
PhD thesis advisor: Tichy Robert, Dr.phil., O.Univ.-Prof.
organization: Working Group Mathematics A of the Institute of Mathematics
year of publication: 2003