Related papers: Minimal Gr\"obner bases and the predictable leadin…

This Paper defines and explores solution to the problem of \emph{Inversion of a finite Sequence} over the binary field, that of finding a prefix element of the sequence which confirms with a \emph{Recurrence Relation} (RR) rule defined by a…

In this article, we investigate the cardinality of Groebner bases under various monomial orderings. We identify a family of polynomials F and a criterion such that the reduced Groebner basis of F is double exponential in cardinality with…

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…

In a previous article (see \cite{CNP}), we introduced and analyzed ring-theoretic properties of object unital $\mathcal{G}$-graded rings $R$, where $\mathcal{G}$ is a groupoid. In the present article, we analyze the category $\grmod$ of…

Flat-injective presentations were introduced by Miller (2020) to provide combinatorial descriptions of $\mathbb Z^n$-graded modules. We consider them in the setting of local graded rings $R$, with grading over an abelian group, and give a…

We define the notion of componentwise regularity and study some of its basic properties. We prove an analogue, when working with weight orders, of Buchberger's criterion to compute Gr\"obner bases; the proof of our criterion relies on a…

We introduce and study the minimum distance function of a graded ideal in a polynomial ring with coefficients in a field, and show that it generalizes the minimum distance of projective Reed-Muller-type codes over finite fields. This gives…

Modular algorithm are widely used in computer algebra systems (CAS), for example to compute efficiently the gcd of multivariate polynomials. It is known to work to compute Groebner basis over $\Q$, but it does not seem to be popular among…

In this article, we study the combinatorics of congruence subgroups of the modular group. More precisely, we consider the notion of minimal monomial solutions. These are the solutions of a matrix equation (also appearing in the study of…

We display a new family of prime ideals with unbounded minimal number of generators in a three-dimensional power series ring over a field of characteristic zero. These primes are obtained as the kernel of a quasi-monomial algebra…

In this article we compute a minimal Groebner basis for the symmetric algebra for certain affine Monomial Curves, as an R-module. Keywords: Monomial Curves, Groebner Basis, Symmetric Algebra. Mathematics Subject Classification 2000: 13P10,…

In this paper, we give a sufficient condition for a set $\mathal G$ of polynomials to be a Gr\"obner basis with respect to a given term-order for the ideal $I$ that it generates. Our criterion depends on the linkage pattern of the ideal $I$…

We develop a new Low-level, First-order Probabilistic Programming Language (LF-PPL) suited for models containing a mix of continuous, discrete, and/or piecewise-continuous variables. The key success of this language and its compilation…

We develop a Gr\"obner basis theory for a class of algebras that generalizes both PBW-algebras and rings of differential algebras on smooth varieties. Emphasis lies on methods to compute filtrations and graded structures defined by weight…

Let K be a field and let M_n(K) denote the space of n x n matrices with entries in K. Let M be a subspace of M_n(K) of dimension d with the property that there are elements in M with non-zero determinant. Given a basis of M, we define the…

Let $P_N(R)$ be the space of all real polynomials in $N$ variables with the usual inner product $<, >$ on it, given by integrating over the unit sphere. We start by deriving an explicit combinatorial formula for the bilinear form…

In this paper we will define analogs of Gr\"obner bases for $R$-subalgebras and their ideals in a polynomial ring $R[x_1,\ldots,x_n]$ where $R$ is a noetherian integral domain with multiplicative identity and in which we can determine ideal…

Partially Observable Markov Decision Processes (POMDPs) model decision making under uncertainty. While there are many approaches to approximately solving POMDPs, we aim to address the problem of learning such models. In particular, we are…

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…

Polynomials and elements over finite fields exhibit closely related algebraic structures, and many properties defined for elements extend naturally to polynomials. The concepts of order and $\mathbb{F}_q$-Order for elements have been…