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## Closed polynomials and saturated subalgebras of polynomials algebras

Ukrainian Mathematical Journal. 2007. Vol. 59. No. 12. P. 1783-1790.
Arzhantsev I., Petravchuk A.

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.