Noether's forms for the study of non-composite rational functions and their spectrum

Laurent Busé 1 Guillaume Chèze 2 Salah Najib 3
1 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : In this paper, the spectrum and the decomposability of a multivariate rational function are studied by means of the effective Noether's irreducibility theorem given by Ruppert. With this approach, some new effective results are obtained. In particular, we show that the reduction modulo p of the spectrum of a given integer multivariate rational function r coincides with the spectrum of the reduction of r modulo p for p a prime integer greater or equal to an explicit bound. This bound is given in terms of the degree, the height and the number of variables of r. With the same strategy, we also study the decomposability of r modulo p. Some similar explicit results are also provided for the case of polynomials with coefficients in a polynomial ring.
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Laurent Busé, Guillaume Chèze, Salah Najib. Noether's forms for the study of non-composite rational functions and their spectrum. Acta Arithmetica, Instytut Matematyczny PAN, 2011, 147 (3), pp.217-231. ⟨10.4064/aa147-3-2⟩. ⟨inria-00395839⟩

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