Related papers: Toric Border Bases
In this paper, we generalize the notion of border bases of zero-dimensional polynomial ideals to the module setting. To this end, we introduce order modules as a generalization of order ideals and module border bases of submodules with…
Border bases arise as a canonical generalization of Gr\"obner bases. We provide a polyhedral characterization of all order ideals (and hence border bases) that are supported by a zero-dimensional ideal: order ideals that support a border…
Border bases are a generalization of Gr\"obner bases for zero-dimensional ideals in polynomial rings. In this article, we introduce border bases for a non-commutative ring of linear differential operators, namely the rational Weyl algebra.…
In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming it complex (resp. real) variety is finite. The aim is to combine approaches for solving a system of polynomial equations with dual…
We introduce an elementary method to study the border rank of polynomials and tensors, analogous to the apolarity lemma. This can be used to describe the border rank of all cases uniformly, including those very special ones that resisted a…
We illustrate an efficient new method for handling polynomial systems with degenerate solution sets. In particular, a corollary of our techniques is a new algorithm to find an isolated point in every excess component of the zero set (over…
Border bases can be considered to be the natural extension of Gr\"obner bases that have several advantages. Unfortunately, to date the classical border basis algorithm relies on (degree-compatible) term orderings and implicitly on reduced…
A relaxation method based on border basis reduction which improves the efficiency of Lasserre's approach is proposed to compute the optimum of a polynomial function on a basic closed semi algebraic set. A new stopping criterion is given to…
We consider linear systems on toric varieties of any dimension, with invariant base points, giving a characterization of special linear systems. We then make a new conjecture for linear systems on rational surfaces.
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…
Toric (or sparse) elimination theory is a framework developped during the last decades to exploit monomial structures in systems of Laurent polynomials. Roughly speaking, this amounts to computing in a \emph{semigroup algebra}, \emph{i.e.}…
Border bases, a generalization of Groebner bases, have actively been researched during recent years due to their applicability to industrial problems. A. Kehrein and M. Kreuzer formulated the so called Border Basis Algorithm, an algorithm…
In this note we discuss an abstract framework for standard boundary value problems in divergence form with maximal monotone relations as "coefficients". A reformulation of the respective problems is constructed such that they turn out to be…
Using basic topology and linear algebra, we define a plethora of invariants of boundary links whose values are power series with noncommuting variables. These turn out to be useful and elementary reformulations of an invariant originally…
Solving systems of polynomial equations, particularly those with finitely many solutions, is a crucial challenge across many scientific fields. Traditional methods like Gr\"obner and Border bases are fundamental but suffer from high…
Matroids generalize the familiar notion of linear dependence from linear algebra. Following a brief discussion of founding work in computability and matroids, we use the techniques of reverse mathematics to determine the logical strength of…
We establish basic information about border rank algorithms for the matrix multiplication tensor and other tensors with symmetry. We prove that border rank algorithms for tensors with symmetry (such as matrix multiplication and the…
Infinite hyperplane arrangements whose vertices form a lattice are studied from the point of view of commutative algebra. The quotient of such an arrangement modulo the lattice action represents the minimal free resolution of the associated…
The main topic of the paper is the construction of various explicit flat families of border bases. To begin with, we cover the punctual Hilbert scheme Hilb^\mu(A^n) by border basis schemes and work out the base changes. This enables us to…
Loop torsors over Laurent polynomial rings in characteristic 0 were originally introduced in relation to infinite dimensional Lie theory. Applications to other areas require a theory that can yields results in positive characteristic, and…