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Irreducible Jacobian derivations in positive characteristic

100%
Jędrzejewicz P.
Open Mathematics
|
2014
|
tom 12
|
nr 8
1278-1284
EN
We prove that an irreducible polynomial derivation in positive characteristic is a Jacobian derivation if and only if there exists an (n-1)-element p-basis of its ring of constants. In the case of two variables we characterize these derivations in terms of their divergence and some nontrivial constants.
2
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Positive characteristic analogs of closed polynomials

76%
Jędrzejewicz P.
Open Mathematics
|
2011
|
tom 9
|
nr 1
50-56
EN
The notion of a closed polynomial over a field of zero characteristic was introduced by Nowicki and Nagata. In this paper we discuss possible ways to define an analog of this notion over fields of positive characteristic. We are mostly interested in conditions of maximality of the algebra generated by a polynomial in a respective family of rings. We also present a modification of the condition of integral closure and discuss a condition involving partial derivatives.
3
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The constants of the Volterra derivation

76%
Hegedűs P.
Open Mathematics
|
2012
|
tom 10
|
nr 3
969-973
EN
The ring of constants of the Volterra derivation is found. Confirming a conjecture of Zielinski, it is always a polynomial ring.
4
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A characterization of p-bases of rings of constants

76%
Jędrzejewicz P.
Open Mathematics
|
2013
|
tom 11
|
nr 5
900-909
EN
We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a unique factorization domain of characteristic p > 0. One of these conditions involves Jacobians while the other some properties of factors. In the case m = n this extends the known theorem of Nousiainen, and we obtain a new formulation of the Jacobian conjecture in positive characteristic.
5
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Border bases and kernels of homomorphisms and of derivations

64%
Zieliński J.
Open Mathematics
|
2010
|
tom 8
|
nr 4
780-785
EN
Border bases are an alternative to Gröbner bases. The former have several more desirable properties. In this paper some constructions and operations on border bases are presented. Namely; the case of a restriction of an ideal to a polynomial ring (in a smaller number of variables), the case of the intersection of two ideals, and the case of the kernel of a homomorphism of polynomial rings. These constructions are applied to the ideal of relations and to factorizable derivations.
6
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The five-variable Volterra system

64%
Zieliński J.
Open Mathematics
|
2011
|
tom 9
|
nr 4
888-896
EN
We give a description of all polynomial constants of the five-variable Volterra derivation, hence of all polynomial first integrals of its corresponding Volterra system of differential equations. The Volterra system plays a significant role in plasma physics and population biology.
7
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Rings of constants of four-variable Lotka-Volterra systems

64%
Zieliński J.
Open Mathematics
|
2013
|
tom 11
|
nr 11
1923-1931
EN
Lotka-Volterra systems appear in population biology, plasma physics, laser physics and derivation theory, among many others. We determine the rings of constants of four-variable Lotka-Volterra derivations with four parameters C 1, C 2, C 3, C 4 ∈ k, where k is a field of characteristic zero. Thus, we give a full description of polynomial first integrals of the respective systems of differential equations.
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