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Closed polynomials and saturated subalgebras of polynomial algebras. (English) Zbl 1164.13302

Ukr. Mat. Zh. 59, No. 12, 1587-1593 (2007) and Ukr. Math. J. 59, No. 12, 1783-1790 (2007).
Summary: The behaviour of closed polynomials, i.e., polynomials \(f\in\mathbb{K}[x_1,\dots,x_n]\setminus\mathbb{K}\) such that the subalgebra \(\mathbb{K}[f]\) is integrally closed in \(\mathbb{K}[x_1,\dots,x_n]\), is studied under extensions of the ground field. Using some properties of closed polynomials, we prove that, after shifting by constants, every polynomial \(f\in\mathbb{K}[x_1,\dots,x_n]\setminus\mathbb{K}\) can be factorized into a product of irreducible polynomials of the same degree. We consider some types of saturated subalgebras \(A\subset\mathbb{K}[x_1,\dots,x_n]\), i.e., subalgebras such that, for any \(f\in A\setminus\mathbb{K}\), a generative polynomial of \(f\) is contained in \(A\).

MSC:

13B25 Polynomials over commutative rings
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