Book contents
- Frontmatter
- Contents
- Preface
- Programme Committee
- Tutorials
- Research Papers
- The Fractal Walk
- Gröbner Bases Property on Elimination Ideal in the Noncommutative Case
- 17 The CoCoA 3 Framework for a Family of Buchberger-like Algorithms
- 18 Newton Identities in the Multivariate Case: Pham Systems
- 19 Gröbner Bases in Rings of Differential Operators
- 20 Canonical Curves and the Petri Scheme
- 21 The Buchberger Algorithm as a Tool for Ideal Theory of Polynomial Rings in Constructive Mathematics
- 22 Gröbner Bases in Non-Commutative Reduction Rings
- 23 Effective Algorithms for Intrinsically Computing SAGBI-Gröbner Bases in a Polynomial Ring over a Field
- 24 De Nugis Groebnerialium 1: Eagon, Northcott, Gröbner
- 25 An application of Gröbner Bases to the Decomposition of Rational Mappings
- 26 On some Basic Applications of Gröbner Bases in Non-commutative Polynomial Rings
- 27 Full Factorial Designs and Distracted Fractions
- 28 Polynomial interpolation of Minimal Degree and Gröbner Bases
- 29 Inversion of Birational Maps with Gröbner Bases
- 30 Reverse Lexicographic Initial Ideals of Generic Ideals are Finitely Generated
- 31 Parallel Computation and Gröbner Bases: An Application for Converting Bases with the Gröbner Walk
- Appendix An Algorithmic Criterion for the Solvability of a System of Algebraic Equations (translated by Michael Abramson and Robert Lumbert)
- Index of Tutorials
30 - Reverse Lexicographic Initial Ideals of Generic Ideals are Finitely Generated
Published online by Cambridge University Press: 05 July 2011
- Frontmatter
- Contents
- Preface
- Programme Committee
- Tutorials
- Research Papers
- The Fractal Walk
- Gröbner Bases Property on Elimination Ideal in the Noncommutative Case
- 17 The CoCoA 3 Framework for a Family of Buchberger-like Algorithms
- 18 Newton Identities in the Multivariate Case: Pham Systems
- 19 Gröbner Bases in Rings of Differential Operators
- 20 Canonical Curves and the Petri Scheme
- 21 The Buchberger Algorithm as a Tool for Ideal Theory of Polynomial Rings in Constructive Mathematics
- 22 Gröbner Bases in Non-Commutative Reduction Rings
- 23 Effective Algorithms for Intrinsically Computing SAGBI-Gröbner Bases in a Polynomial Ring over a Field
- 24 De Nugis Groebnerialium 1: Eagon, Northcott, Gröbner
- 25 An application of Gröbner Bases to the Decomposition of Rational Mappings
- 26 On some Basic Applications of Gröbner Bases in Non-commutative Polynomial Rings
- 27 Full Factorial Designs and Distracted Fractions
- 28 Polynomial interpolation of Minimal Degree and Gröbner Bases
- 29 Inversion of Birational Maps with Gröbner Bases
- 30 Reverse Lexicographic Initial Ideals of Generic Ideals are Finitely Generated
- 31 Parallel Computation and Gröbner Bases: An Application for Converting Bases with the Gröbner Walk
- Appendix An Algorithmic Criterion for the Solvability of a System of Algebraic Equations (translated by Michael Abramson and Robert Lumbert)
- Index of Tutorials
Summary
Abstract
This article generalizes the well-known notion of generic forms to the algebra R′, introduced in [12]. For the total degree, then reverse lexicographic order, we prove that the initial ideal of an ideal generated by finitely many generic forms (in countably infinitely many variables) is finitely generated. This contrasts to the lexicographic order, for which initial ideals of generic ideals in general are non-finitely generated.
The natural question, “is the reverse lexicographic initial ideal of an homogeneous, finitely generated ideal in R′ finitely generated” is posed, but not answered; we do, however, point out one direction of investigation that might provide the answer: namely to view such an ideal as the “specialization” of a generic ideal.
Introduction
In this article, we study the initial ideals of generic and “almost generic” ideals with respect to the (total degree, then) reverse lexicographic term order. For a generic ideal I ⊂ K[x1,…, xn], generated by r ≤ n forms, there is a well-known conjecture on how gr(I) looks like. In particular, gr(I) is minimally generated in K[x1,…, xr]. We interpret this result in the setting of the ring R′, introduced in [12]: this ring, which is a proper subring on the power series ring on countably many variables, and which properly contains the polynomial ring on the same set of indeterminates, is the habitat of “generic forms in countably many indeterminates”.
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- Gröbner Bases and Applications , pp. 504 - 518Publisher: Cambridge University PressPrint publication year: 1998