On certain algebraic curves related to polynomial maps (vol 103, pg 319, 1996)

摘要

An argument is given to fill a gap in a proof in the author's article On certain algebraic curves related to polynomial maps, Compositio Math. 103 (1996), 319-350, that the polynomial Phi(n)(x, c), whose roots are the periodic points of period n of a certain polynomial map x -> f (x, c), is absolutely irreducible over the finite field of p elements, provided that f (x, 1) has distinct roots and that the multipliers of the orbits of period n are also distinct over F(p). Assuming that Phi(n)(x, c) is reducible in characteristic p, we show that Hensel's lemma and Laurent series expansions of the roots can be used to obtain a factorization of Phi(n)(x, c) in characteristic 0, contradicting the absolute irreducibility of this polynomial over the rational field