Estimating the number of tetrahedra determined by volume, circumradius and four face areas using Groebner basis

Tsai, Ya-Lun

doi:10.1016/j.jsc.2016.02.002

Cited by 6 publications

(3 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Gröbner bases have enjoyed a diverse set of applications since their inception in 1965 (for example, see [12,13,16,17]). In 2004, Gröbner bases were applied to the problem of model selection in systems biology [11].…”

Section: Introductionmentioning

confidence: 99%

Small Gröbner fans of ideals of points

Dimitrova

Robbiano

et al. 2019

J. Algebra Appl.

View full text Add to dashboard Cite

In the context of modeling biological systems, it is of interest to generate ideals of points with a unique reduced Gröbner basis, and the first main goal of this paper is to identify classes of ideals in polynomial rings which share this property. Moreover, we provide methodologies for constructing such ideals. We then relax the condition of uniqueness. The second and most relevant topic discussed here is to consider and identify pairs of ideals with the same number of reduced Gröbner bases, that is, with the same cardinality of their associated Gröbner fan.the space of PDSs that fit the data and a minimal model is selected from the space by computing a reduced Gröbner basis of the ideal and taking the normal forms of the model equations. While this provides an algorithmic solution to model selection, each choice of monomial order results in a different minimal PDS, with each one yielding different hypotheses about the underlying biological network. The following example illustrates this claim.Lactose metabolism in E.coli is controlled by the lac operon, a genetic system made up of simultaneously transcribed genes. It is said that the lac operon (x) is ON (lactose is metabolized) when the activating protein CAP (y) is present and when the inhibiting protein lacI (z) is absent. This behavior can be described by the Boolean function f = y ∧ ¬z; as a polynomial over the finite field F 2 , we can write f = y(z + 1) = yz + y. If we consider the inputs X = {(1, 0, 0), (0, 1, 0), (1, 0, 1)} representing Boolean states for the lac operon, CAP, and lacI respectively, then the ideal of polynomials vanishing on X has two Gröbner bases, namely {x 2 + x, z 2 + z, y + x + 1, xz + z} and {y 2 + y, z 2

show abstract

Section: Introductionmentioning

confidence: 99%

Small Gröbner fans of ideals of points

Dimitrova

Robbiano

et al. 2019

J. Algebra Appl.

View full text Add to dashboard Cite

show abstract

“…In 1965, Bruno Buchberger introduced Gröbner bases, which are multivariate nonlinear generalizations of echelon forms [4,5]. Since this landmark thesis, the adoption of Gröbner bases has expanded into diverse fields, such as geometry [21], imagine processing [15], oil industry [20], quantum field theory [17], and systems biology [14].…”

Section: Introductionmentioning

confidence: 99%

The Number of Gröbner Bases in Finite Fields

Zhang,

Stigler

2019

Preprint

View full text Add to dashboard Cite

In the field of algebraic systems biology, the number of minimal polynomial models constructed using discretized data from an underlying system is related to the number of distinct reduced Gröbner bases for the ideal of the data points. While the theory of Gröbner bases is extensive, what is missing is a closed form for their number for a given ideal. This work contributes connections between the geometry of data points and the number of Gröbner bases associated to small data sets. Furthermore we improve an existing upper bound for the number of Gröbner bases specialized for data over a finite field.

show abstract

“…Gröbner bases have enjoyed a diverse set of applications since their inception in 1965 (for example, see [1,2,3,4]). However, some applications have found a challenge in the fact that a polynomial ideal can have multiple reduced Gröbner bases, thus yielding importance to the question of characterizing ideals of points with unique reduced Gröbner bases.…”

Section: Introductionmentioning

confidence: 99%

Geometric characterization of data sets with unique reduced Gröbner bases

Dimitrova,

He,

Stigler

et al. 2018

Preprint

View full text Add to dashboard Cite

Applications of Gröbner bases, such as reverse engineering of gene regulatory networks and combinatorial encoding of receptive fields, consider data sets whose ideals of points have unique reduced Gröbner bases. The significance is that uniqueness provides a canonical representation of the input data. In this work, we identify geometric properties of input data that result in a unique reduced Gröbner basis. We show that if the data form a staircase or a so-called linear shift of a staircase, the ideal of the points has a unique reduced Gröbner basis. These results serve to minimize computational effort in using Gröbner bases and are a stepping stone for developing algorithms to generate such data sets.

show abstract

Estimating the number of tetrahedra determined by volume, circumradius and four face areas using Groebner basis

Cited by 6 publications

References 1 publication

Small Gröbner fans of ideals of points

Small Gröbner fans of ideals of points

The Number of Gröbner Bases in Finite Fields

Geometric characterization of data sets with unique reduced Gröbner bases

Contact Info

Product

Resources

About