Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Number of Grobner Bases in Finite Fields
53
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 Grobner bases for the ideal of the data points. While the theory of Grobner 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 Grobner bases associated to small data sets. Furthermore we improve an existing upper bound for the number of Grobner bases specialized for data over a finite field.
rate research
Read More
Model selection based on experimental data is an important challenge in biological data science. Particularly when collecting data is expensive or time consuming, as it is often the case with clinical trial and biomolecular experiments, the problem of selecting information-rich data becomes crucial for creating relevant models. We identify geometric properties of input data that result in a unique algebraic model and 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 Gro bner basis and thus corresponds to a unique model. We use linear shifts to partition data into equivalence classes with the same basis. We demonstrate the utility of the results by applying them to a Boolean model of the well-studied lac operon in E. coli.
In this paper, we make a contribution to the computation of Grobner bases. For polynomial reduction, instead of choosing the leading monomial of a polynomial as the monomial with respect to which the reduction process is carried out, we investigate what happens if we make that choice arbitrarily. It turns out not only this is possible (the fact that this produces a normal form being already known in the literature), but, for a fixed choice of reductors, the obtained normal form is the same no matter the order in which we reduce the monomials. To prove this, we introduce reduction machines, which work by reducing each monomial independently and then collecting the result. We show that such a machine can simulate any such reduction. We then discuss different implementations of these machines. Some of these implementations address inherent inefficiencies in reduction machines (repeating the same computations). We describe a first implementation and look at some experimental results.
In this work we provide a definition of a coloured operad as a monoid in some monoidal category, and develop the machinery of Grobner bases for coloured operads. Among the examples for which we show the existance of a quadratic Grobner basis we consider the seminal Lie-Rinehart operad whose algebras include pairs (functions, vector fields).
Grassmann manifolds $G_{k,n}$ are among the central objects in geometry and topology. The Borel picture of the mod 2 cohomology of $G_{k,n}$ is given as a polynomial algebra modulo a certain ideal $I_{k,n}$. The purpose of this paper is to understand this cohomology via Grobner bases. Reduced Grobner bases for the ideals $I_{k,n}$ are determined. An application of these bases is given by proving an immersion theorem for Grassmann manifolds $G_{5,n}$, which establishes new immersions for an infinite family of these manifolds.
We give a notion of combinatorial proximity among strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. We show that this notion guarantees geometric proximity of the corresponding points in the Hilbert scheme. We define a graph whose vertices correspond to strongly stable ideals and whose edges correspond to pairs of adjacent ideals. Every term order induces an orientation of the edges of the graph. This directed graph describes the behavior of the points of the Hilbert scheme under Grobner degenerations with respect to the given term order. Then, we introduce a polyhedral fan that we call Grobner fan of the Hilbert scheme. Each cone of maximal dimension corresponds to a different directed graph induced by a term order. This fan encodes several properties of the Hilbert scheme. We use these tools to present a new proof of the connectedness of the Hilbert scheme. Finally, we improve the technique introduced in the paper Double-generic initial ideal and Hilbert scheme by Bertone, Cioffi and Roggero to give a lower bound on the number of irreducible components of the Hilbert scheme.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
