On the computation of graded components of Laurent polynomial rings

2006/05/21 by Sonia L. Rueda, Rueda, Sonia L.
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.math/0605567

openalex publication_date 2006/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we present several algorithms for dealing with graded components of Laurent polynomial rings. To be more precise, let S be the Laurent polynomial ring k[x₁,...,x_r,x_r+1^{± 1},..., x_n^{± 1}], k algebraicaly closed field of characteristic 0. We define the multigrading of S by an arbitrary finitely generated abelian group A. We construct a set of fans compatible with the multigrading and use this fans to compute the graded components of S using polytopes. We give an algorithm to check whether the graded components of S are finite dimensional. Regardless of the dimension, we determine a finite set of generators of each graded component as a module over the component of homogeneous polynomials of degree 0.

Related

• Laurent cancellation for rings of transcendence degree one
• Graded Linearity of Stanley-Reisner Ring of Broken Circuit Complexes
• Computing Implicitizations of Multi-Graded Polynomial Maps
• ℤ-graded identities of the Lie algebras U₁
• The Grothendieck Ring of Certain Non-Noetherian Group-Graded Algebras via K-Series