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1
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Border bases and kernels of homomorphisms and of derivations

100%
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.
2
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Rational Constants of Generic LV Derivations and of Monomial Derivations

100%
Zieliński J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2013
|
tom 61
|
nr 3
201-208
EN
We describe the fields of rational constants of generic four-variable Lotka-Volterra derivations. Thus, we determine all rational first integrals of the corresponding systems of differential equations. Such systems play a role in population biology, laser physics and plasma physics. They are also an important part of derivation theory, since they are factorizable derivations. Moreover, we determine the fields of rational constants of a class of monomial derivations.
3
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On the algebra of constants of polynomial derivations in two variables

100%
Zieliński J.
Colloquium Mathematicum
|
2000
|
tom 83
|
nr 2
267-269
EN
Let d be a k-derivation of k[x,y], where k is a field of characteristic zero. Denote by $\widetilde d$ the unique extension of d to k(x,y). We prove that if ker d ≠ k, then ker $\widetilde d$ = (ker d)_0, where (ker d)_0 is the field of fractions of ker d.
4
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The five-variable Volterra system

100%
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.
5
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Local derivations in polynomial and power series rings

100%
Zieliński J.
Colloquium Mathematicum
|
2002
|
tom 92
|
nr 2
295-305
EN
We give a description of all local derivations (in the Kadison sense) in the polynomial ring in one variable in characteristic two. Moreover, we describe all local derivations in the power series ring in one variable in any characteristic.
6
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Rings of constants of four-variable Lotka-Volterra systems

100%
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.
7
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Polynomial algebra of constants of the four variable Lotka-Volterra system

64%
Ossowski P., Zieliński J.
Colloquium Mathematicum
|
2010
|
tom 120
|
nr 2
299-309
EN
We describe the ring of constants of a specific four variable Lotka-Volterra derivation. We investigate the existence of polynomial constants in the other cases of Lotka-Volterra derivations, also in n variables.
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