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Closed and Irreducible Polynomials in Several Variables

Ivan V. Arzhantsev·Anatoliy P. Petravchuk
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Abstract

New and old results on closed polynomials, i.e., such polynomials f in K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n], are collected. Using some properties of closed polynomials we prove the following factorization theorem: Let f be an element of K[x_1,...,x_n], where K is an algebraically closed field. Then for all but finite number of a's the polynomial f+a can be decomposed into a product of irreducible polynomials of the same degree (not depending on a) such that their defferences are constants. An algorithm for finding of a generative polynomial of a given polynomial f, which is a closed polynomial h with f=F(h) for some F(t) in K[t], is given. Some types of saturated subalgebras A in K[x_1,...,x_n] are considered, i.e., such that for any f in A a generative polynomial of f is contained in A.

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Ivan V. Arzhantsev, Anatoliy P. Petravchuk. 2007-05-20. Closed and Irreducible Polynomials in Several Variables. https://arxiv.org/abs/math/0608157

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