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.
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References
M. Ayad, “Sur les polynômes f(X,Y) tels que K[f] est intégralement fermé dans K[X,Y],” Acta Arithm., 105, No. 1, 9–28 (2002).
A. Nowicki, “On the Jacobian equation J(f,g) = 0 for polynomials in k[x,y],”, Nagoya Math. J., 109, 151–157 (1988).
A. Nowicki and M. Nagata, “Rings of constants for k-derivations in k[x 1, …, x n],” J. Math. Kyoto Univ., 28, 111–118 (1988).
A. Schinzel, Polynomials with Special Regard to Reducibility, Cambridge University Press, Cambridge (2000).
N. Bourbaki, Elements of Mathematics, Commutative Algebra, Springer, Berlin (1989).
I. R. Shafarevich, Basic Algebraic Geometry I, Springer, Berlin (1994).
A. Zaks, “Dedekind subrings of k[x 1, …, x n] are rings of polynomials,” Isr. J. Math., 9, 285–289 (1971).
Y. Stein, “The total reducibility order of a polynomial in two variables,” Isr. J. Math., 68, 109–122 (1989).
E. Cygan, “Factorization of polynomials,” Bull. Polish. Acad. Sci. Math., 40, 45–52 (1992).
D. Lorenzini, “Reducibility of polynomials in two variables,” J. Algebra, 156, 65–75 (1993).
S. Kaliman, “Two remarks on polynomials in two variables,” Pacif. J. Math., 154, 285–295 (1992).
A. Vistoli, “The number of reducible hypersurfaces in a pencil,” Invent. Math., 112, 247–262 (1993).
S. Najib, “Une généralisation de l’inégalité de Stein-Lorenzini,” J. Algebra, 292, 566–573 (2005).
W. Fulton, “Introduction to toric varieties,” Ann. Math. Stud., 131 (1993).
S. Lang, Algebra, Springer (2002).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1587–1593, December, 2007.
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Arzhantsev, I.V., Petravchuk, A.P. Closed polynomials and saturated subalgebras of polynomial algebras. Ukr Math J 59, 1783–1790 (2007). https://doi.org/10.1007/s11253-008-0037-4
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DOI: https://doi.org/10.1007/s11253-008-0037-4