Closed polynomials and saturated subalgebras of polynomial algebras
The behavior of closed polynomials, i.e., polynomials $f ∈ k[x_1,…,x_n]∖k$ such that the subalgebra $k[f]$ is integrally closed in $k[x_1,…,x_n]$, is studied under extensions of the ground field. Using some properties of closed polynomials, we prove that, after shifting by constants, every polynomial $f ∈ k[x_1,…,x_n]∖k$ can be factorized into a product of irreducible polynomials of the same degree. We consider some types of saturated subalgebras $A ⊂ k[x_1,…,x_n]$, i.e., subalgebras such that, for any $f ∈ A∖k$, a generative polynomial of $f$ is contained in $A$.
English version (Springer): Ukrainian Mathematical Journal 59 (2007), no. 12, pp 1783-1790.
Citation Example: Arzhantsev I. V., Petravchuk A. P. Closed polynomials and saturated subalgebras of polynomial algebras // Ukr. Mat. Zh. - 2007. - 59, № 12. - pp. 1587–1593.