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We address some questions concerning indecomposable polynomials and their spectrum. How does the spectrum behave via reduction or specialisation, or via a more general ring morphism? Are the indecomposability properties equivalent over a field and over its algebraic closure? How many polynomials are decomposable over a finite field?
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 in . 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… (More)
Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that we fully describe, if one coefficient is allowed to vary, then the polynomial is irreducible for all but at most deg(P ) − 1 values of the coefficient. We more generally handle the situation where several specified coefficients vary.