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1
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On rings of constants of derivations in two variables in positive characteristic

100%
Jędrzejewicz P.
Colloquium Mathematicum
|
2006
|
tom 106
|
nr 1
109-117
EN
Let k be a field of chracteristic p > 0. We describe all derivations of the polynomial algebra k[x,y], homogeneous with respect to a given weight vector, in particular all monomial derivations, with the ring of constants of the form $k[x^{p},y^{p},f]$, where $f ∈ k[x,y]∖ k[x^{p},y^{p}]$.
2
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Positive characteristic analogs of closed polynomials

100%
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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A Characterization of One-Element p-Bases of Rings of Constants

100%
Jędrzejewicz P.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2011
|
tom 59
|
nr 1
19-26
EN
Let K be a unique factorization domain of characteristic p > 0, and let f ∈ K[x₁,...,xₙ] be a polynomial not lying in $K[x₁^p,...,xₙ^p]$. We prove that $K[x₁^p,...,xₙ^p,f]$ is the ring of constants of a K-derivation of K[x₁,...,xₙ] if and only if all the partial derivatives of f 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.
4
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Linear derivations with rings of constants generated by linear forms

100%
Jędrzejewicz P.
Colloquium Mathematicum
|
2008
|
tom 113
|
nr 2
279-286
EN
Let k be a field. We describe all linear derivations d of the polynomial algebra k[x₁,...,xₘ] such that the algebra of constants with respect to d is generated by linear forms: (a) over k in the case of char k = 0, (b) over $k[x₁^{p},...,xₘ^{p}]$ in the case of char k = p > 0.
5
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Quasi-commutative polynomial algebras and the power property of 2 × 2 quantum matrices

100%
Jędrzejewicz P.
Colloquium Mathematicum
|
1996
|
tom 71
|
nr 2
217-224
6
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A note on characterizations of rings of constants with respect to derivations

100%
Jędrzejewicz P.
Colloquium Mathematicum
|
2004
|
tom 99
|
nr 1
51-53
EN
Let A be a commutative algebra without zero divisors over a field k. If A is finitely generated over k, then there exist well known characterizations of all k-subalgebras of A which are rings of constants with respect to k-derivations of A. We show that these characterizations are not valid in the case when the algebra A is not finitely generated over k.
7
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A note on rings of constants of derivations in integral domains

100%
Jędrzejewicz P.
Colloquium Mathematicum
|
2011
|
tom 122
|
nr 2
241-245
EN
We observe that the characterization of rings of constants of derivations in characteristic zero as algebraically closed subrings also holds in positive characteristic after some natural adaptation. We also present a characterization of such rings in terms of maximality in some families of rings.
8
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Linear gradings of polynomial algebras

100%
Jędrzejewicz P.
Open Mathematics
|
2008
|
tom 6
|
nr 1
13-24
EN
Let k be a field, let $$ G $$ be a finite group. We describe linear $$ G $$-gradings of the polynomial algebra k[x 1, ..., x m] such that the unit component is a polynomial k-algebra.
9
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A characterization of p-bases of rings of constants

100%
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.
10
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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.
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