Closed and Irreducible Polynomials in Several Variables

Details Closed and Irreducible Polynomials in Several Variables Closed and Irreducible Polynomials in Several Variables

2006-08-07

arxiv.org/abs/math/0608157v2

Ukrain. Mat. Zh. 59:12 (2007), 1587--1593; transl. in Ukrainian Math. J. 59:12 (2007), 1783--1790

Mathematics

Commutative Algebra

13 pages, changes in Section 4

Scientific paper

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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