4 Local Gröbner Fan : Polyhedral and Computational Approach
نویسنده
چکیده
The goal of this paper is to show that the local Gröbner fan is a polyhedral fan for ideals in the ring of power series and the homogenized ring of analytic differential operators. We will also discuss about relations between the local Gröbner fan and the (global) Gröbner fan for a given ideal and algorithms of computing local Gröbner fans. In rings of differential operators, finiteness and convexity of local Gröbner fans were firstly proved by Assi, Castro-Jiménez and Granger. But they did not prove that they are polyhedral fans.
منابع مشابه
Local Gröbner fan
This note summarizes results concerning local Gröbner fans. Any Gröbner fan of an ideal in a power series ring, or in an analytic or formal differential operator ring, is found to be a polyhedral fan. We also compare the notions of global and local Gröbner fans, and discuss applications of our results. To cite this article: R. Bahloul, N. Takayama, C. R. Acad. Sci. Paris, Ser. I 340 (2005).
متن کاملA Non-Regular Grobner Fan
The Gröbner fan of an ideal I ⊂ k[x1, . . . , xn], defined by Mora and Robbiano, is a complex of polyhedral cones in Rn. The maximal cones of the fan are in bijection with the distinct monomial initial ideals of I as the term order varies. If I is homogeneous the Gröbner fan is complete and is the normal fan of the state polytope of I. In general the Gröbner fan is not complete and therefore no...
متن کاملBOOKS THAT NEED REVIEWERS FOR THE SIGACT NEWS COLUMN Algorithms 1. ReCombinatorics: The algorithmics of ancestral recombination graphs and phylogenic networks by Gusfield. 2. Distributed Systems: An algorithmic approach (second edition) by Ghosh. 3. Tractability: Practical approach to Hard Problems
4. Polyhedral and Algebraic Methods in Computational Geometry, by Michael Joswig and Thorsten Theobald. Reviewed by Brittany Terese Fasy and David L. Millman. This book treats a wide array of topics, beginning from the basics (e.g., convex hulls) and proceeding through the fundamental tools of computational algebraic geometry (e.g., Gröbner bases), and containing a section on applications as well.
متن کاملComputing Gröbner fans
This paper presents algorithms for computing the Gröbner fan of an arbitrary polynomial ideal. The computation involves enumeration of all reduced Gröbner bases of the ideal. Our algorithms are based on a uniform definition of the Gröbner fan that applies to both homogeneous and non-homogeneous ideals and a proof that this object is a polyhedral complex. We show that the cells of a Gröbner fan ...
متن کاملGröbner fans of x-homogeneous ideals in R〚t〛[x]
We generalise the notion of Gröbner fan to ideals in RJtK[x1, . . . , xn] for certain classes of coefficient rings R and give a constructive proof that the Gröbner fan is a rational polyhedral fan. For this we introduce the notion of initially reduced standard bases and show how these can be computed in finite time. We deduce algorithms for computing the Gröbner fan, implemented in the computer...
متن کامل