Related papers: On Gr\"obner Basis Detection for Zero-dimensional …
We generalize signature Gr\"obner bases, previously studied in the free algebra over a field or polynomial rings over a ring, to ideals in the mixed algebra $R[x_1,...,x_k]\langle y_1,\dots,y_n \rangle$ where $R$ is a principal ideal…
This paper describes and analyzes a method for computing border bases of a zero-dimensional ideal $I$. The criterion used in the computation involves specific commutation polynomials and leads to an algorithm and an implementation extending…
There are several efficient methods to solve linear interval polynomial systems in the context of interval computations, however, the general case of interval polynomial systems is not yet covered as well. In this paper we introduce a new…
Solving polynomial systems arising from applications is frequently made easier by the structure of the systems. Weighted homogeneity (or quasi-homogeneity) is one example of such a structure: given a system of weights…
The main idea in this paper is merging two techniques that have been recently developed. On the one hand, we consider MCCGS, standing for Minimal Canonical Comprehensive Groebner Systems, a recently introduced computational tool yielding…
Signature-based algorithms are the latest and most efficient approach as of today to compute Gr\"obner bases for polynomial systems over fields. Recently, possible extensions of these techniques to general rings have attracted the attention…
Previous work by Mora and Sala provides the reduced Groebner basis of the ideal formed by the elementary symmetric polynomials in $n$ variables of degrees $k=1,\dots,n$, $\langle e_{1,n}(x), \dots, e_{n,n}(x) \rangle$. Haglund, Rhoades, and…
Triangular decomposition with different properties has been used for various types of problem solving, e.g. geometry theorem proving, real solution isolation of zero-dimensional polynomial systems, etc. In this paper, the concepts of strong…
Experiences with the implementation of strong Gr\"obner bases respectively standard bases for polynomial rings over principal ideal rings are explained: different strategies for creating the pair set, methods to avoid coefficient growth and…
The purpose of this paper is twofold. An immediate practical use of the presented algorithm is its applicability to the parametric solution of underdetermined linear ordinary differential equations (ODEs) with coefficients that are…
Let $(f\_1,\dots, f\_s) \in \mathbb{Q}\_p [X\_1,\dots, X\_n]^s$ be a sequence of homogeneous polynomials with $p$-adic coefficients. Such system may happen, for example, in arithmetic geometry. Yet, since $\mathbb{Q}\_p$ is not an effective…
We introduce a very natural topology on the set of total orderings of monomials of any algebra having a countable basis over a field. This topological space and some notable subspaces are compact. This topological framework allows us to…
We consider the problem of mutually unbiased bases as a polynomial optimization problem over the reals. We heavily reduce it using known symmetries before exploring it using two methods, combining a number of optimization techniques. The…
In [4] Sturmfels linked the Hilbert Nullstellensatz to Gr\"obner bases through final polynomials. In (loc. cit.) it was claimed that final polynomials always appear in a lexicographic Gr\"obner basis of a certain ideal. In this paper, we…
We establish Gr\"{o}bner-Shirshov bases theory for Gelfand-Dorfman-Novikov algebras over a field of characteristic $0$. As applications, a PBW type theorem in Shirshov form is given and we provide an algorithm for solving the word problem…
The complexity of computing the solutions of a system of multivariate polynomial equations by means of Groebner bases computations is upper bounded by a function of the solving degree. In this paper, we discuss how to rigorously estimate…
A set of polynomials G in a polynomial ring S over a field is said to be a universal Groebner basis, if G is a Groebner basis with respect to every term order on S. Twenty years ago Bernstein, Sturmfels, and Zelevinsky proved that the set…
Let $I_1\subset I_2\subset\dots$ be an increasing sequence of ideals of the ring $\Bbb Z[X]$, $X=(x_1,\dots,x_n)$ and let $I$ be their union. We propose an algorithm to compute the Gr\"obner base of $I$ under the assumption that the…
Nakajima's graded quiver varieties naturally appear in the study of bases of cluster algebras. One particular family of these varieties, namely the bipartite determinantal varieties, can be defined for any bipartite quiver and gives a vast…
It is proposed the algorithm that find a basis of the ideal and a basis of the space of all root functionals by using the extension operation for bounded root functionals, when the number of polynomials is equal to the number of variables,…