Related papers: On modular computation of Groebner bases with inte…

We develop the theory of Gr\"obner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-one correspondences among various sets of partitions by using division…

In this paper we present an algorithm for computing Groebner bases of linear ideals in a difference polynomial ring over a ground difference field. The input difference polynomials generating the ideal are also assumed to be linear. The…

This extended abstract gives a construction for lifting a Gr\"obner basis algorithm for an ideal in a polynomial ring over a commutative ring R under the condition that R also admits a Gr\"obner basis for every ideal in R.

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…

Although Buchberger's algorithm, in theory, allows us to compute Gr\"obner bases over any field, in practice, however, the computational efficiency depends on the arithmetic of the ground field. Consider a field $K = \mathbb{Q}(\alpha)$, a…

In this paper we present a new efficient variant to compute strong Gr\"obner basis over quotients of principal ideal domains. We show an easy lifting process which allows us to reduce one computation over the quotient $R/nR$ to two…

In this article we present two new algorithms to compute the Groebner basis of an ideal that is invariant under certain permutations of the ring variables and which are both implemented in SINGULAR (cf. [DGPS12]). The first and major…

Standard noncommutative Gr\"obner basis procedures are used for computing ideals of free noncommutative polynomial rings over fields. This paper describes Gr\"obner basis procedures for one-sided ideals in finitely presented noncommutative…

In this paper we compute Gr\"obner bases for determinantal ideals of the form $I_{1}(XY)$, where $X$ and $Y$ are both matrices whose entries are indeterminates over a field $K$. We use the Gr\"obner basis structure to determine Betti…

Let K be a field with a valuation and let S be the polynomial ring S:= K[x_1,..., x_n]. We discuss the extension of Groebner theory to ideals in S, taking the valuations of coefficients into account, and describe the Buchberger algorithm in…

Assuming sufficiently many terms of a n-dimensional table defined over a field are given, we aim at guessing the linear recurrence relations with either constant or polynomial coefficients they satisfy. In many applications, the table terms…

In this paper we describe an efficient involutive algorithm for constructing Groebner bases of polynomial ideals. The algorithm is based on the concept of involutive monomial division which restricts the conventional division in a certain…

Let $\Lambda$ be a commutative Noetherian ring, and let $I$ be a proper ideal of $\Lambda$, $R=\Lambda /I$. Consider the polynomial rings $T=\Lambda [x_1,...x_n]$ and $A=R[x_1,...,x_n]$. Suppose that linear equations are solvable in…

Algorithmic computation in polynomial rings is a classical topic in mathematics. However, little attention has been given to the case of rings with an infinite number of variables until recently when theoretical efforts have made possible…

We study an inductive method of computing initial ideals and Gr\"obner bases for families of ideals in a polynomial ring. This method starts from a given set of pairs $(I,J)$ where $I$ is any ideal and $J$ is a monomial ideal contained in…

Let $I = ( f_1, \dots, f_n )$ be a homogeneous ideal in the polynomial ring $K[x_1, \dots,x_n]$ over a field $K$ generated by generic polynomials. Using an incremental approach based on a method by Gao, Guan and Volny, and properties of the…

We define a new type of ideal basis called the proper basis that improves both Gr\"obner basis and Buchberger's algorithm. Let $x_1$ be the least variable of a monomial ordering in a polynomial ring $K[x_1,\dotsc,x_n]$ over a field $K$. The…

The universal Gr\"obner basis of an ideal is a Gr\"obner basis with respect to all term orders simultaneously. The aim of this paper is to present an algorithmic approach to compute the universal Gr\"obner basis for the toric ideal…

Classically, Groebner bases are computed by first prescribing a set monomial order. Moss Sweedler suggested an alternative and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build…

A \textit{symmetric ideal} $I \subseteq R = K[x_1,x_2,...]$ is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Gr\"obner bases for symmetric ideals in the infinite…