On implementing the symbolic preprocessing function over Boolean polynomial rings in Gröbner basis algorithms using linear algebra

2015 ◽  
Vol 29 (3) ◽  
pp. 789-804 ◽  
Author(s):  
Yao Sun ◽  
Zhenyu Huang ◽  
Dongdai Lin ◽  
Dingkang Wang
Keyword(s):  
Linear Algebra ◽  
Gröbner Basis ◽  
Polynomial Rings ◽  
Grobner Basis

scholarly journals Subalgebra Analogue to standard basis for ideal

2011 ◽  
Vol 48 (4) ◽  
pp. 458-474
Author(s):  
Junaid Khan
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Standard Basis ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
Grobner Basis

A theory of “subalgebra basis” analogous to standard basis (the generalization of Gröbner bases to monomial orderings which are not necessarily well orderings [1]) for ideals in polynomial rings over a field is developed. We call these bases “SASBI Basis” for “Subalgebra Analogue to Standard Basis for Ideals”. The case of global orderings, here they are called “SAGBI Basis” for “Subalgebra Analogue to Gröbner Basis for Ideals”, is treated in [6]. Sasbi bases may be infinite. In this paper we consider subalgebras admitting a finite Sasbi basis and give algorithms to compute them.

An FGLM-like algorithm for computing the radical of a zero-dimensional ideal

2018 ◽  
Vol 17 (01) ◽  
pp. 1850002 ◽  
Author(s):  
Teo Mora
Keyword(s):  
Linear Algebra ◽  
Linear Form ◽  
Gröbner Basis ◽  
Good Representation ◽  
Grobner Basis

We present a linear algebra algorithm, which, given a zero-dimensional ideal via a “good” representation, produces a separating linear form, its radical, again given via the same “good” representation and also a Kronecker parametrization of it. As an irrelevant byproduct, if you like, you can also read a Gröbner basis of the radical.

Synergy in the Theories of Gröbner Bases and Path Algebras

1993 ◽  
Vol 45 (4) ◽  
pp. 727-739 ◽  
Author(s):  
Daniel R. Farkas ◽  
C. D. Feustel ◽  
Edward L. Green
Keyword(s):  
General Theory ◽  
Gröbner Basis ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
Grobner Basis ◽  
Associative Algebras ◽  
Path Algebras

AbstractA general theory for Grôbner basis in path algebras is introduced which extends the known theory for commutative polynomial rings and free associative algebras.

Balancing sets of vectors

2010 ◽  
Vol 47 (3) ◽  
pp. 333-349 ◽  
Author(s):  
Gábor Hegedűs
Keyword(s):  
Lower Bound ◽  
Linear Algebra ◽  
Scalar Product ◽  
Gröbner Basis ◽  
Prime Factor ◽  
Grobner Basis ◽  
Arbitrary Integer ◽  
Usual Scalar ◽  
Usual Scalar Product ◽  
Minimal Integer

Let n be an arbitrary integer, let p be a prime factor of n . Denote by ω1 the pth primitive unity root, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\omega _1 : = e^{\tfrac{{2\pi i}} {p}}$$ \end{document}.Define ωi ≔ ω1i for 0 ≦ i ≦ p − 1 and B ≔ {1, ω1 , …, ωp −1 } n ⊆ ℂ n .Denote by K ( n; p ) the minimum k for which there exist vectors ν1 , …, νk ∈ B such that for any vector w ∈ B , there is an i , 1 ≦ i ≦ k , such that νi · w = 0, where ν · w is the usual scalar product of ν and w .Gröbner basis methods and linear algebra proof gives the lower bound K ( n; p ) ≧ n ( p − 1).Galvin posed the following problem: Let m = m ( n ) denote the minimal integer such that there exists subsets A1 , …, Am of {1, …, 4 n } with | Ai | = 2 n for each 1 ≦ i ≦ n , such that for any subset B ⊆ [4 n ] with 2 n elements there is at least one i , 1 ≦ i ≦ m , with Ai ∩ B having n elements. We obtain here the result m ( p ) ≧ p in the case of p > 3 primes.

scholarly journals Small Gröbner fans of ideals of points

2019 ◽  
Vol 19 (05) ◽  
pp. 2050087 ◽  
Author(s):  
Elena Dimitrova ◽  
Qijun He ◽  
Lorenzo Robbiano ◽  
Brandilyn Stigler
Keyword(s):  
Gröbner Basis ◽  
Biological Systems ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
Grobner Basis ◽  
Relevant Topic ◽  
Gröbner Fan ◽  
Grobner Fan

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.

scholarly journals An Algorithm to Compute the H-Bases for Ideals of Subalgebras

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Rabia ◽  
Muhammad Ahsan Binyamin ◽  
Nazia Jabeen ◽  
Adnan Aslam ◽  
Kraidi Anoh Yannick
Keyword(s):  
Polynomial Ring ◽  
Gröbner Basis ◽  
Polynomial Rings ◽  
Grobner Basis ◽  
Finitely Generated ◽  
Computational Algebra ◽  
Important Ingredient ◽  
Multivariate Polynomial

The concept of H-bases, introduced long ago by Macauly, has become an important ingredient for the treatment of various problems in computational algebra. The concept of H-bases is for ideals in polynomial rings, which allows an investigation of multivariate polynomial spaces degree by degree. Similarly, we have the analogue of H-bases for subalgebras, termed as SH-bases. In this paper, we present an analogue of H-bases for finitely generated ideals in a given subalgebra of a polynomial ring, and we call them “HSG-bases.” We present their connection to the SAGBI-Gröbner basis concept, characterize HSG-basis, and show how to construct them.

scholarly journals On the first fall degree of summation polynomials

2019 ◽  
Vol 13 (3-4) ◽  
pp. 229-237
Author(s):  
Stavros Kousidis ◽  
Andreas Wiemers
Keyword(s):  
Elliptic Curve ◽  
Gröbner Basis ◽  
Discrete Logarithm ◽  
Polynomial Systems ◽  
Discrete Logarithm Problem ◽  
Grobner Basis ◽  
Weil Descent ◽  
Degree Bound

Abstract We improve on the first fall degree bound of polynomial systems that arise from a Weil descent along Semaev’s summation polynomials relevant to the solution of the Elliptic Curve Discrete Logarithm Problem via Gröbner basis algorithms.

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