✦ LIBER ✦

Reduced Gröbner Bases Under Composition

✍ Scribed by J. Gutierrez; R.R. San Miguel

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 463 KB
Volume: 26
Category: Article
ISSN: 0747-7171
DOI: 10.1006/jsco.1998.0222

No coin nor oath required. For personal study only.

✦ Synopsis

In this paper we contribute with one main result to the interesting problem initiated by Hong (1998, J. Symb. Comput. 25, 643-663) on the behaviour of Gröbner bases under composition of polynomials. Polynomial composition is the operation of replacing the variables of a polynomial with other polynomials. The main question of this paper is: When does composition commute with reduced Gröbner bases computation under the same term ordering? We give a complete answer for this question: let Θ be a polynomial map, then for every reduced Gröbner basis G, G • Θ is a reduced Gröbner basis if and only if the composition by Θ is compatible with the term ordering and Θ is a list of permuted univariate and monic polynomials. Besides, we also include other minor results concerned with this problem; in particular, we provide a sufficient condition to determine when composition commutes with reduced Gröbner bases computation (possibly) under different term ordering.

📜 SIMILAR VOLUMES

Multiplicative Bases, Gröbner Bases, and

Multiplicative Bases, Gröbner Bases, and Right Gröbner Bases

✍ Edward L. Green 📂 Article 📅 2000 🏛 Elsevier Science 🌐 English ⚖ 314 KB

In this paper, we study conditions on algebras with multiplicative bases so that there is a Gröbner basis theory. We introduce right Gröbner bases for a class of modules. We give an elimination theory and intersection theory for right submodules of projective modules in path algebras. Solutions to h

Regular Gröbner Bases

✍ Jonas MÅnsson; Patrik Nordbeck 📂 Article 📅 2002 🏛 Elsevier Science 🌐 English ⚖ 366 KB

In this paper we introduce the concept of bi-automaton algebras, generalizing the automaton algebras previously defined by Ufnarovski. A bi-automaton algebra is a quotient of the free algebra, defined by a binomial ideal admitting a Gröbner basis which can be encoded as a regular set; we call such a

On the Stability of Gröbner Bases Under

On the Stability of Gröbner Bases Under Specializations

✍ MICHAEL KALKBRENER 📂 Article 📅 1997 🏛 Elsevier Science 🌐 English ⚖ 344 KB

Let R be a Noetherian commutative ring with identity, K a field and π a ring homomorphism from R to K. We investigate for which ideals in R[x 1 , . . . , xn] and admissible orders the formation of leading monomial ideals commutes with the homomorphism π.

Counting and Gröbner Bases

✍ K. Kalorkoti 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 239 KB

We show how the complexity of counting relates to the well known phenomenon that computing Gröbner bases under a lexicographic order is generally harder than total degree orders. We give simple examples of polynomials for which it is very easy to compute their Gröbner basis using a total degree orde

Gröbner Bases and Polyhedral Geometry of

Gröbner Bases and Polyhedral Geometry of Reducible and Cyclic Models

✍ Serkan Hoşten; Seth Sullivant 📂 Article 📅 2002 🏛 Elsevier Science 🌐 English ⚖ 234 KB

This article studies the polyhedral structure and combinatorics of polytopes that arise from hierarchical models in statistics, and shows how to construct Gro¨bner bases of toric ideals associated to a subset of such models. We study the polytopes for cyclic models, and we give a complete polyhedral

Converting Bases with the Gröbner Walk

✍ S. COLLART; M. KALKBRENER; D. MALL 📂 Article 📅 1997 🏛 Elsevier Science 🌐 English ⚖ 299 KB

We present an algorithm which converts a given Gröbner basis of a polynomial ideal I to a Gröbner basis of I with respect to another term order. The conversion is done in several steps following a path in the Gröbner fan of I. Each conversion step is based on the computation of a Gröbner basis of a