Polynomial solutions of the Diophantine equation a(x^p-y^q)=b(z^r-w^s), with 1/p+1/q+1/r+1/s=1 and related results.
Event details
| Date | 19.12.2012 |
| Hour | 11:15 › 12:30 |
| Speaker | Maciej Ulas (Krakow) |
| Location | |
| Category | Conferences - Seminars |
In this talk we will present some results related to the existence of polynomial solutions of the Diophantine equation a(x^p-y^q)=b(z^r-w^s), where a, b are given non-zero integers and the exponents satisfy the condition 1/p+1/q+1/r+1/s=1. In particular, for any quadruple (p,q,r,s) such that all entries are even and not all equal to 4 there are infinitely many polynomial solutions (defined over Z). In case of quadruplets (2, 4, 6, 12), (2, 6, 6, 6), (2, 4, 8, 8), (2, 8, 4, 8) we present constructions of primitive polynomial solutions, i. e. polynomial solutions which are co-prime. We show that in each case the set of rational points on the underlying surface is dense in the Zariski topology. For the surface with (p,q,r,s)=(2,6,6,6) we prove density of rational points in the Euclidean topology. At the end of the talk we give some generalization of the presented results for higher dimensional varieties of the form a(x^4- P(X)^2)=b(y^4-Q(X)^2), where X is a vector of n variables and P, Q are homogenous forms. This is joint work with Andrew Bremner (Arizona State University).
Practical information
- General public
- Free
Organizer
- CIB
Contact
- Isabelle Derivaz-Rabii