Computational Linear and Commutative Algebra

Kreuzer, Martin; Robbiano, Lorenzo

doi:10.1007/978-3-319-43601-2

Cited by 85 publications

(197 citation statements)

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“…As one can read in the work of Kreuzer and Robbiano (2005): ''Boo Barkee revealed his love for mathematics when he licked a draft of the paper about SAGBI'' (p. 477). According to Levy-dit-Vehel, Marinari, Perret, and Traverso (2009), some attribute legendary roots to Boo Barkee, stating that he was a French general of Greek origin (why do they hook onto French generals so much?).…”

Section: Family Connectionsmentioning

confidence: 99%

Mathematicians Who Never Were

Pieronkiewicz

2018

Math Intelligencer

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Section: Family Connectionsmentioning

confidence: 99%

Mathematicians Who Never Were

Pieronkiewicz

2018

Math Intelligencer

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“…If G is a Gröbner basis, then H R/ G = H R/ lt(G) , and it is easy to compute the related Hilbert series or Hilbert polynomial for H R/ G (d) from lt (G) [2][3] [27]. Many textbooks, such as [23], contain further details.…”

Section: 32mentioning

confidence: 99%

Reducing the size and number of linear programs in a dynamic Gröbner basis algorithm

Caboara

Perry

2014

AAECC

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Abstract. The dynamic algorithm to compute a Gröbner basis is nearly twenty years old, yet it seems to have arrived stillborn; aside from two initial publications, there have been no published followups. One reason for this may be that, at first glance, the added overhead seems to outweigh the benefit; the algorithm must solve many linear programs with many linear constraints. This paper describes two methods that reduce both the size and number of these linear programs.

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“…In order to carry out explicit calculations with Theorem 4, one has to work out a Gröbner basis of the ideal J or a border basis of J [5]. A computational alternative to this theorem is the elimination theory (see e.g., [6]), however Theorem 4 is much more attractive aesthetically.…”

Section: Propositionmentioning

confidence: 99%

An Eigenvalue Theorem for Systems of Polynomial Equations

Billig

Dixon²

2014

The American Mathematical Monthly

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We give a new short proof of a theorem relating solutions of a system of polynomial equations to the eigenvalues of the multiplication operators on the quotient ring, in the case when the quotient ring is …nite-dimensional.Everyone is familiar with the representation of a curve in the plane using an algebraic equation such as X 2 + Y 2 = 1. Algebraic geometers have learned that it is more convenient to represent these curves in the equivalent form of solutions (or zeros) of polynomials. In the case above, the circle is simply the set of ( ; ) which are zeros of X 2 + Y 2 1. Similarly the simultaneous zeros of several polynomials represent the points of intersection of all of the corresponding curves. The solution of a system of polynomial equations is a natural extension of the problem of solving linear equations, and arises, for example, in the Lagrange multiplier method when the constraints and the function to be optimized are algebraic. Our particular aim is to prove the theorem below (Theorem 4) which gives a description of the common zeros of a system of polynomials. It is interesting in itself because it combines various important concepts and results from a standard undergraduate curriculum: ideals, quotient rings, homomorphism theorems, commuting linear operators and their common eigenvectors. Although the theorem is not new (see [2], [4] and [3]) and is quite elementary, we cannot …nd it in undergraduate text books.If K is any …eld and f 1 ; : : : ; f n are polynomials in the ring K[X; Y ], then it is convenient to consider the ideal J = hf 1 ; : : : ; f n i generated by these polynomials. The set of common zeros in K 2 of f 1 ; : : : ; f n is clearly the same as the set of common zeros for all the polynomials in J so we can forget about the particular polynomials chosen to generate J and simply think about the ideal J itself. We shall refer to these zeros brie ‡y as the zeros of J. An important advantage of approaching the problem of the set of common zeros of polynomials (= intersection of curves) in this way is that we can take advantage of the structure of the ring K[X; Y ] rather than simply dealing with a subset of K 2 . At the same time, we should not lose sight of the geometric interpretation of the theorems which arise.In what follows, we shall restrict ourselves to the case of two variables, but at the end of this note we shall point out how all the results can be generalized to the case of m variables X 1 ; : : : ; X m .

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Computational Linear and Commutative Algebra

Cited by 85 publications

References 1 publication

Mathematicians Who Never Were

Mathematicians Who Never Were

Reducing the size and number of linear programs in a dynamic Gröbner basis algorithm

An Eigenvalue Theorem for Systems of Polynomial Equations

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