Noether forms for the study of non-composite rational functions and their spectrum
Laurent Busé, Guillaume Chèze, Salah Najib · Acta Arithmetica · 2011
Introduction.Consider a non-constant polynomial f ∈ K[X 1 , . . ., X n ], n ≥ 2, where K is a field.Denoting by K the algebraic closure of K, the spectrum of f is the setIt is customary to say that f is non-composite if it cannot be written in the form u(h(X)) with h(X) ∈ K[X], u(t) ∈ K[t] and deg(u) ≥ 2. A famous theorem of Bertini states that f is non-composite if and only if σ(f ) is finite; see for instance [20, Theorem 37].Furthermore, Stein proved in [22] that if f is non-composite, then the cardinality of σ(f ) does not exceed deg(f ) -1; see also [17,16,7,12].Recently in [4], A. Bodin, P. Dèbes, and S. Najib have studied the behavior of the spectrum of a polynomial via a ring morphism.Here we generalize this study to the spectrum of a rational function and we give explicit bounds.Let f and g be two non-constant relatively prime polynomials in K[X 1 , . . ., X n ], n ≥ 2. The spectrum of the rational function r = f /g ∈ K(X 1 , . . ., X n ) is the set 2010