Displaying similar documents to “Computations with Frobenius powers.”

Border bases and kernels of homomorphisms and of derivations

Janusz Zieliński (2010)

Open Mathematics

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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. ...

Reduction of Power Series in a Polydisc with Respect to a Gröbner Basis

Justyna Szpond (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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We deal with a reduction of power series convergent in a polydisc with respect to a Gröbner basis of a polynomial ideal. The results are applied to proving that a Nash function whose graph is algebraic in a "large enough" polydisc, must be a polynomial. Moreover, we give an effective method for finding this polydisc.

A characterization of p-bases of rings of constants

Piotr Jędrzejewicz (2013)

Open Mathematics

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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.

An algorithm for primary decomposition in polynomial rings over the integers

Gerhard Pfister, Afshan Sadiq, Stefan Steidel (2011)

Open Mathematics

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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. ...

Solving linear systems of equations over integers with Gröbner bases

Amir Hashemi (2014)

Acta Arithmetica

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We introduce a novel application of Gröbner bases to solve (non-homogeneous) systems of integer linear equations over integers. For this purpose, we present a new algorithm which ascertains whether a linear system of equations has an integer solution or not; in the affirmative case, the general integer solution of the system is determined.