Derivations and Radicals of Polynomial Ideals over Fields of Arbitrary Characteristic

Fortuna, Elisabetta; Gianni, Patrizia; Trager, Barry M.

doi:10.1006/jsco.2002.0525

Cited by 7 publications

(4 citation statements)

References 14 publications

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“…This is a classical, standard problem in symbolic computation. We refer to [13,15,17] for the related work. So, we do not focus on how to compute generators of I(Y ) in this paper.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Symmetric Tensor Decompositions On Varieties

Nie¹,

Yao²,

Zhi³

2020

Preprint

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This paper discusses the problem of symmetric tensor decomposition on a given variety X: decomposing a symmetric tensor into the sum of tensor powers of vectors contained in X. In this paper, we first study geometric and algebraic properties of such decomposable tensors, which are crucial to the practical computations of such decompositions. For a given tensor, we also develop a criterion for the existence of a symmetric decomposition on X. Secondly and most importantly, we propose a method for computing symmetric tensor decompositions on an arbitrary X. As a specific application, Vandermonde decompositions for nonsymmetric tensors can be computed by the proposed algorithm.

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“…This is a classical, standard problem in symbolic computation. We refer to [13,15,17] for the related work. So, we do not focus on how to compute generators of I(Y ) in this paper.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Symmetric Tensor Decompositions On Varieties

Nie¹,

Yao²,

Zhi³

2020

Preprint

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“…The problem of finding generators of radical ideals in polynomial rings is addressed in [40], from a computational point of view.…”

Section: Definition 14mentioning

confidence: 99%

A constructive notion of codimension

Rinaldi

2013

Journal of Algebra

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We give a point-free and constructively meaningful characterization of the notion of codimension for ideals of a commutative ring, which with classical logic and the axiom of choice is equivalent to the customary one. This characterization is based on notions from formal topology, and upon the elementary characterization of Krull dimension provided earlier by Coquand, Lombardi, and Roy. Among other things, we use this notion to prove Krull's principal ideal theorem in the context of constructive algebra.

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“…The first radical ideal algorithms assumed characteristic 0. However, in recent publications algorithms have been suggested that compute radical ideals over base fields which are finitely generated over a finite field or over Q (see [11,16,19]).…”

Section: Algebraic Groupsmentioning

confidence: 99%

Quantum automata and algebraic groups

Derksen

Jeandel

Koiran

2005

Journal of Symbolic Computation

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We show that several problems which are known to be undecidable for probabilistic automata become decidable for quantum finite automata. Our main tool is an algebraic result of independent interest: we give an algorithm which, given a finite number of invertible matrices, computes the Zariski closure of the group generated by these matrices. RésuméNous montrons ici que plusieurs problèmes indécidables pour des automates probabilistes sont décidables pour des automates quantiques. Ce rsultat s'appuie sur un algorithme intéressant en soi, qui,étant donné des matrices inversibles, calcule la cloture de Zariski du groupe engendré par ses matrices.Abstract. We show that several problems which are known to be undecidable for probabilistic automata become decidable for quantum finite automata. Our main tool is an algebraic result of independent interest: we give an algorithm which, given a finite number of invertible matrices, computes the Zariski closure of the group generated by these matrices.Harm Derksen is partially supported by NSF, grant DMS 0102193.

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Derivations and Radicals of Polynomial Ideals over Fields of Arbitrary Characteristic

Cited by 7 publications

References 14 publications

Symmetric Tensor Decompositions On Varieties

Symmetric Tensor Decompositions On Varieties

A constructive notion of codimension

Quantum automata and algebraic groups

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