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Ukrainian Mathematical Journal

, 59:1783 | Cite as

Closed polynomials and saturated subalgebras of polynomial algebras

  • I. V. Arzhantsev
  • A. P. Petravchuk
Article

Abstract

The behavior of closed polynomials, i.e., polynomials \( f \in \Bbbk [x_1 , \ldots ,x_n ]\backslash \Bbbk \) such that the subalgebra \( \Bbbk [f] \) is integrally closed in \( \Bbbk [x_1 , \ldots ,x_n ] \) , is studied under extensions of the ground field. Using some properties of closed polynomials, we prove that, after shifting by constants, every polynomial \( f \in \Bbbk [x_1 , \ldots ,x_n ]\backslash \Bbbk \) can be factorized into a product of irreducible polynomials of the same degree. We consider some types of saturated subalgebras \( A \subset \Bbbk [x_1 , \ldots ,x_n ] \) , i.e., subalgebras such that, for any \( f \in A\backslash \Bbbk \) , a generative polynomial of f is contained in A.

Keywords

Generative Element Generative Polynomial Irreducible Polynomial Polynomial Algebra Integral Closure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • I. V. Arzhantsev
    • 1
  • A. P. Petravchuk
    • 2
  1. 1.Moscow State UniversityMoscowRussia
  2. 2.Shevchenko Kiev National UniversityKievUkraine

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