Abstract : The purpose of this mini-symposium is to share recent developments in the theory of Gr\"obner bases and their applications. Gr\"obner bases have been studied by many researchers and have been used in various fields including commutative algebra, algebraic geometry, and engineering. Solving interesting open problems and devising efficient algorithms are still highly desired. In this mini-symposium, we will discuss, in particular, the complexity of Gr\"obner basis computation, applications of Gr\"obner bases, and algorithms for Gr\"obner bases in parametric, non-commutative, or valuation polynomial rings.
Abstract : We present several algorithms for parametric ideal operations of polynomial ideals. Let $K[X]=K[x_1,\ldots,x_n]$ be the polynomial ring over a filed $K$ and $K[A,X]$ the parametric polynomial ring with parameters $A=\{a_1,\ldots,a_m\}$. Let $\varphi_\alpha$ be the homomorphism from $K[A,X]$ to $K[X]$ by $\varphi_\alpha(f(A,X))=f(\alpha,X)$ for $\alpha\in K^m$. For an ideal operation $F$ and parametric ideals $I_1,…,I_r$, we compute a set of pairs $\{(A_i,G_i)\}_{i=1}^s$ s.t. $\bigcup_{i=1}^s A_i=K^m$ and $\varphi_\alpha(G_i)$ is a Groebner basis of $F(\varphi_\alpha(I_1),…,\varphi_\alpha(l_r))$ for any $\alpha\in A_i$.
[01892] On the complexity of Groebner basis computation
Format : Talk at Waseda University
Author(s) :
Kazuhiro Yokoyama (Rikkyo University)
Abstract : The complexity of computation of Groebner basis is heavily related to the syzygy of given generating set. We will discuss this relation for generic case, where coefficients of generating set can be considered as parameters.
[01851] On Parametric Border Basis and Comprehensive Gröbner System
Format : Talk at Waseda University
Author(s) :
Yosuke Sato (Tokyo University of Science)
Abstract : A border basis is an alternative tool to a Gröbner basis
for handling a polynomial ideal.
One of the most important properties of a border basis is its stability.
Though this property is cited by many researchers,
its precise definition has not been given anywhere yet.
We give its rigorous definition in terms of a parametric
polynomial ideal and introduce several properties which enable
us to have a simple representation of a parametric polynomial ideal.
Abstract : Tate algebras are defined as multivariate formal power series over a $p$-adic field, with a convergence condition.
We will present the concept of Universal Analytic Gröbner Basis for a polynomial ideal:
a finite polynomial basis of an ideal such that it is a Gröbner basis in any of the completions into Tate algebras
for any (rational) convergence radius.
We will present how to compute it and its relation with tropical varieties.
[01891] Criteria for Grobner bases and degenerations by structure of signatures
Format : Talk at Waseda University
Author(s) :
Yuta Kambe (Mitsubishi Electric Corporation)
Abstract : The F5 algorithm was presented by Faugere in 2002 and variants of the F5 algorithm have been proposed by various researchers called signature based algorithms (SBAs). The concept of signatures was revised by Arri-Perry in 2011 as normal monomials for syzygies and Vaccon-Yokoyama presented an implementable SBA in 2017 with the concept of guessed signatures. The speaker will talk about these new concepts and his new criterion theorem for Grobner bases and degenerations.
[01876] On signature-based algorithm for tropical Groebner bases on Weyl algebra
Format : Talk at Waseda University
Author(s) :
Ari Dwi Hartanto (Department of Mathematics, Universitas Gadjah Mada)
Katsuyoshi Ohara (Faculty of Mathematics and Physics, Kanazawa University)
Abstract : The computational aspect of tropical Groebner basis for polynomial rings introduced by A.W.Chan is extended to the Weyl algebras over fields with valuations. A term order with valuation is designed to be a generalization of the tropical term orders studied by A.W.Chan and by T.Vaccon. Although it is not well-ordering, a signature-based algorithm can still be developed. The minimal natural signature of a polynomial exists. The F5 criterion is then adopted for this context.
[01898] Algorithms for bivariate lexicographic Groebner bases
Format : Talk at Waseda University
Author(s) :
Xavier DAHAN (Tohoku UniversityTohoku University)
Abstract : The lexicographic monomial order for Groebner bases is fundamental since it holds the elimination property. This also implies strong structural properties, as shown by Lazard in 1985 in the case of two variables. Yet these properties have not been fully exploited. I will explain how they allow to perform Chinese Remainder Theorem like operations in several situations, as well as the difficulties in remaining cases. I will also discuss extensions to more than two variables.
[01885] An algebraic approach to factor analysis
Format : Talk at Waseda University
Author(s) :
Ryoya Fukasaku (Kyushu University)
Kei Hirose (Kyushu University)
Yutaro Kabata (Nagasaki University)
Keisuke Teramoto (Hiroshima University)
Abstract : When the maximum likelihood method is used in factor analysis, it is not uncommon for an unique variance less than or equal to zero to be generated, which is known as the improper solution problem. Although numerical approaches have been made to this problem, algebraic approaches have not. Therefore, we aim to exactly describe the solution space associated with the maximum likelihood method in factor analysis by making use of computational algebraic methods such as Gröbner bases and cylindrical algebraic decomposition.
