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On the size of sets in a polynomial variant of a problem of Diophantus

In the poster I will present one polynomial variant of the problem of Diophantus, described in the paper A. Dujella and A. Jurasic, On the size of sets in a polynomial variant of a problem of Diophantus, Int. J. Number Theory 6 (2010), 1449-1471. The problem of Diophantus is to find Diophantine m-tuples, sets of m positive integers with the property that the product of any two of its distinct elements plus 1 is a perfect square. In the article, we considered the problem over K[X], for an algebraically closed field K of characteristic 0. The main result was that there does not exist such set of 8 polynomials, not all constant, with coe±cients in K with the property that the product of any two of its distinct elements plus 1 is a perfect square. This is an improvement of the previously known bound of 11 polynomials. We got an improvement of an upper bound for the size of a set in K[X] with the property that, for a given n in Z[X], the prod- uct of any two of its distinct elements plus 1 is a pure power. We also proved that in K[X] the conjecture that for every Diophantine quadruple {; ; ; a ; b ; c ; d}; ; ; we have (a + b - c - d)2 = 4(ab + 1)(cd + 1), which is true in Z[X], does not hold.

Diophantine m-tuples ; polynomials ; function ¯elds ; Ramsey theory

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