A characterization of one-element p-bases of rings of constants

• Autorzy: Jędrzejewicz P.
• Języki publikacji: EN

Abstrakty

EN

Let Κ be a unique factorization domain of characteristic p > 0, and let ƒ ∈ Κ[χ₁,..., χ_n] be a polynomial not lying in Κ[wzór]. We prove that Κ[wzór] is the ring of constants of a Κ-derivation of Κ[χ₁,..., χ_n] if and only if all the partial derivatives of ƒ are relatively prime. The proof is based on a generalization of Freudenburg's lemma to the case of polynomials over a unique factorization domain of arbitrary characteristic.

Słowa kluczowe

EN

derivation; ring of constants; p-basis

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2011
• Tom: Vol. 59, no 1
• Strony: 19--26
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Jędrzejewicz P.

• Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, 87-100 Toruń, Poland,

Bibliografia

• [1] D. Daigle, Locally nilpotent derivations, lecture notes for the September School of Algebraic Geometry (Łukęcin, 2003) (unpublished), aixl.uottawa.ca/~ddaigle/.
• [2] A. van den Essen, A. Nowicki and A. Tyc, Generalizations of a lemma of Freudenburg, J. Pure Appl. Algebra 177 (2003), 43-47.
• [3] G. Freudenburg, A note on the kernel of a locally nilpotent derivation, Proc. Amer. Math. Soc. 124 (1996), 27-29.
• [4] P. Jędrzejewicz, A note on rings of constants of derivations in integral domains, Colloq. Math. 122 (2011), 241-245.
• [5] P. Jędrzejewicz, One-element p-bases of rings of constants of derivations, Osaka J. Math. 46 (2009), 223-234.
• [6] P. Jędrzejewicz, Rings of constants of p-homogeneous polynomial derivations, Comm. Algebra 31 (2003), 5501-5511.
• [7] H. Matsumura, Commutative Algebra, 2nd ed., Benjamin, Reading, MA, 1980.
• [8] A. Nowicki, Polynomial derivations and their rings of constants, Nicolaus Coperni-cus University, Torun, 1994, www.mat.umk.pl/~anow/.
• [9] A. Nowicki, Rings and fields of constants for derivations in characteristic zero, J. Pure Appl. Algebra 96 (1994), 47-55.
• [10] A. Nowicki and M. Nagata, Rings of constants for k-derivations in k[x1,..., xn], J. Math. Kyoto Univ. 28 (1988), 111-118.