Closed and Irreducible Polynomials in Several Variables

New and old results on closed polynomials, i.e., such polynomials f ∈ k[x1, . . . , xn]\k that the subalgebra k[f ] is integrally closed in k[x1, . . . , xn], are collected in the paper. Using some properties of closed polynomials we prove the following factorization theorem: Let f ∈ k[x1, . . . , xn] \ k, where k is algebraically closed. Then for all but finite number μ ∈ k the polynomial f + μ can be decomposed into a product f + μ = α · f1μ · f2μ · · · fkμ, α ∈ k ×, k > 1, of irreducible polynomials fiμ of the same degree d not depending on μ and such that fiμ − fjμ ∈ k, i, j = 1, . . . , k. 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) ∈ k[t], is given. Some types of saturated subalgebras A ⊂ k[x1, . . . , xn] are considered, i.e., such that for any f ∈ A\k a generative polynomial of f is contained in A.