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1
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An algorithm for primary decomposition in polynomial rings over the integers

100%
Pfister G., Sadiq A., Steidel S.
Open Mathematics
|
2011
|
tom 9
|
nr 4
897-904
EN
We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals, resp. over finite fields, and the idea of Shimoyama-Yokoyama, resp. Eisenbud-Hunecke-Vasconcelos, to extract primary ideals from pseudo-primary ideals. A parallelized version of the algorithm is implemented in Singular. Examples and timings are given at the end of the article.
2
Content available remote

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.
3
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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.
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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A procedure to compute prime filtration

64%
Rauf A.
Open Mathematics
|
2010
|
tom 8
|
nr 1
26-31
EN
Let K be a field, S = K[x 1, … x n] be a polynomial ring in n variables over K and I ⊂ S be an ideal. We give a procedure to compute a prime filtration of S/I. We proceed as in the classical case by constructing an ascending chain of ideals of S starting from I and ending at S. The procedure of this paper is developed and has been implemented in the computer algebra system Singular.
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