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SUMMARY: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.
DTSTART:20121219T111500
DTEND:20121219T123000
DTSTAMP:20260916T061317Z
UID:edec2852bddd7827a61a1214bc4059940bf776b5a06cfa4a32e562e6
CATEGORIES:Conferences - Seminars
DESCRIPTION:Maciej Ulas (Krakow)\nIn this talk we will present some result
 s related to the existence of polynomial solutions of the Diophantine equa
 tion a(x^p-y^q)=b(z^r-w^s)\, where a\, b are given non-zero integers and t
 he 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 equa
 l 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 solutio
 ns\, i. e. polynomial solutions which are co-prime. We show that in each c
 ase the set of rational points on the underlying surface is dense in the Z
 ariski topology. For the surface with (p\,q\,r\,s)=(2\,6\,6\,6) we prove d
 ensity of rational points in the Euclidean topology. At the end of the tal
 k we give some generalization of the presented results for higher dimensio
 nal varieties of the form a(x^4- P(X)^2)=b(y^4-Q(X)^2)\, where X is a vect
 or of n variables and P\, Q are homogenous forms. This is joint work with 
 Andrew Bremner (Arizona State University).
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 nem
STATUS:CONFIRMED
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