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Universal Hilbert subsets

Published online by Cambridge University Press:  01 July 1998

PIERRE DÈBES  and
UMBERTO ZANNIER
PIERRE DÈBES
Affiliation:
Mathématiques, Université Lille 1, 59655 Villeneuve d'Ascq Cedex, France; e-mail: [email protected]
UMBERTO ZANNIER
Affiliation:
Ist. Univ. Arch. D.C.A., Santa Croce, 191, 30135 Venezia, Italy; e-mail: [email protected]
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Abstract

We show that the sequence 2n+n is a universal Hilbert sequence. That is, for each polynomial P(T, Y) irreducible in ℚ(T) [Y], the polynomial P(2n+n, Y) is irreducible in ℚ[Y] for all but finitely many n. This answers a question of M. Yasumoto. Other examples, like 2n+5n, are given. They are all obtained as special cases of a more general result which is proved from classical diophantine arguments.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 124 , Issue 1 , July 1998 , pp. 127 - 134
Copyright
Cambridge Philosophical Society 1998

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