A theorem on refining division orders by the reverse
              lexicographic order

Bayer, David; Stillman, Michael

doi:10.1215/s0012-7094-87-05517-7

Cited by 104 publications

(53 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…. , ζ r ], that is, we obtain only the one elimination ideal [5,19]. For our application, we only need one particular elimination ideal, that given by using BS (O(k)), while the Bayer and Stillman orderings are more efficient than the lexicographic orderings [5].…”

Section: Example 221mentioning

confidence: 99%

Algorithms for the Nonclassical Method of Symmetry Reductions

Clarkson¹,

Mansfield²

1994

SIAM J. Appl. Math.

119

113

View full text Add to dashboard Cite

In this article we present first an algorithm for calculating the determining equations associated with so-called "nonclassical method " of symmetry reductions (à la Bluman and Cole) for systems of partial differential equations. This algorithm requires significantly less computation time than that standardly used, and avoids many of the difficulties commonly encountered. The proof of correctness of the algorithm is a simple application of the theory of Gröbner bases.In the second part we demonstrate some algorithms which may be used to analyse, and often to solve, the resulting systems of overdetermined nonlinear pdes. We take as our principal example a generalised Boussinesq equation, which arises in shallow water theory. Although the equation appears to be non-integrable, we obtain an exact "two-soliton" solution from a nonclassical reduction. *

show abstract

Section: Example 221mentioning

confidence: 99%

Algorithms for the Nonclassical Method of Symmetry Reductions

Clarkson¹,

Mansfield²

1994

SIAM J. Appl. Math.

119

113

View full text Add to dashboard Cite

show abstract

“…The generic initial ideal (gin) of an ideal I of a polynomial ring captures a number of important features of I. When computed with respect to the reverse lexicographic monomial order, gin rev (I) retains two of the most significant measures for the homological complexity of I: the projective dimension and Castelnuovo-Mumford regularity, as shown by Bayer and Stillman in [4]. These groundbreaking results have subsequently been refined and generalized by Bayer-Charalambous-Popescu [2] and Aramova-Herzog [1] to show that gin rev (I) retains also the extremal Betti numbers of I; see Example 1.3 for an illustration of this notion.…”

Section: Introductionmentioning

confidence: 99%

“…Throughout this paper we assume that k is an infinite field, which is a necessary condition for generic initial ideals to be well-defined. The remarkable notion of generic initial ideals was introduced by Galligo in characteristic zero [12] and Bayer and Stillman in arbitrary characteristic [4].…”

Section: Introductionmentioning

confidence: 99%

Axial constants and sectional regularity of homogeneous ideals

DeBellevue¹,

Lebovitz²,

Li³

et al. 2021

Preprint

View full text Add to dashboard Cite

A notion of sectional regularity for a homogeneous ideal I, which measures the regularity of its generic sections with respect to linear spaces of various dimensions, is introduced. It is related to axial constants defined as the intercepts on the coordinate axes of the set of exponents of monomials in the reverse lexicographic generic initial ideal of I. The equivalence of these notions and several other homological and ideal-theoretic invariants is shown. It is also established that these equivalent invariants grow linearly for the family of powers of a given ideal.

show abstract

“…Этот мономиальный порядок был введен Маколеем в [4]. Разница между однородными лексикографическим и обратным лексикографическим порядком довольно невелика, однако использование обратного лексикографического порядка вместо однородного лексикографического приводит к значительному повышению эффективности вычислений во многих алгоритмах [5], [6].…”

unclassified

Идеалы, Порожденные Обратно-Лексикографическими Сегментами

Крупи¹,

Crupi²,

Барбьера³

et al. 2011

Mat. Zametki

View full text Add to dashboard Cite

Пусть-поле, и пусть = [ 1,. .. , ]-кольцо многочленов от переменных 1,. .. , с коэффициентами в поле. Мы рассматриваем идеалы в , порожденные обратно-лексикографическими сегментами мономов из. Идеал, порожденный обратно-лексикографическим сегментом, называется вполне обратно-лексикографически сегментным идеалом, если всевозможные итерированные тени его множества образующих суть обратно-лексикографические сегменты. Мы описываем все вполне обратно-лексикографически сегментные идеалы в и выясняем, при каких условиях у них имеются линейные резольвенты. Библиография: 11 названий.

show abstract

A theorem on refining division orders by the reverse lexicographic order

Cited by 104 publications

References 6 publications

Algorithms for the Nonclassical Method of Symmetry Reductions

Algorithms for the Nonclassical Method of Symmetry Reductions

Axial constants and sectional regularity of homogeneous ideals

Идеалы, Порожденные Обратно-Лексикографическими Сегментами

Contact Info

Product

Resources

About