Related papers: Toric Border Bases
Sparse polynomial systems with vertical coefficient dependencies arise naturally when describing the critical points of optimization problems and, when augmented with linear forms, the steady states of chemical reaction networks. Moreover,…
Given the toric (or toral) arrangement defined by a root system $\Phi$, we describe the poset of its layers (connected components of intersections) and we count its elements. Indeed we show how to reduce to zero-dimensional layers, and in…
Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate…
The theory of border bases for zero-dimensional ideals has attracted several researchers in symbolic computation due to their numerical stability and mathematical elegance. As shown in (Francis & Dukkipati, J. Symb. Comp., 2014), one can…
In this paper, the concepts of binomial difference ideals and toric difference varieties are defined and their properties are proved. Two canonical representations for Laurent binomial difference ideals are given using the reduced Groebner…
We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions…
The solution of constrained linear partial-differential equations can be described via parametric representations of linear relations. To study these representations, we provide a novel definition of boundary triplets for linear relations…
We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool…
Border complexity measures are defined via limits (or topological closures), so that any function which can approximated arbitrarily closely by low complexity functions itself has low border complexity. Debordering is the task of proving an…
In this paper, we extend the idea of comprehensive Gr\"{o}bner bases given by Weispfenning (1992) to border bases for zero dimensional parametric polynomial ideals. For this, we introduce a notion of comprehensive border bases and border…
We generalize Loewner's method for proving that matrix monotone functions are operator monotone. The relation x \leq y on bounded operators is our model for a definition for C*-relations of being residually finite dimensional. Our main…
We present a homological characterisation of those chain complexes of modules over a Laurent polynomial ring in several indeterminates which are finitely dominated over the ground ring (that is, are a retract up to homotopy of a bounded…
We present an explicit solution of the $A_r$ $T$-system for arbitrary boundary conditions. For each boundary, this is done by constructing a network, i.e. a graph with positively weighted edges, and the solution is expressed as the…
We investigate border ranks of twisted powers of polynomials and smoothability of symmetric powers of algebras. We prove that the latter are smoothable. For the former, we obtain upper bounds for the border rank in general and prove that…
In this paper, the concept of toric difference varieties is defined and four equivalent descriptions for toric difference varieties are presented in terms of difference rational parametrization, difference coordinate rings, toric difference…
We present two effective tools for computing the positive tropicalization of algebraic varieties. First, we outline conditions under which the initial ideal can be used to compute the positive tropicalization, offering a real analogue to…
The main goal of the paper is the discussion of a deeper interaction between matrix theory over polynomial rings over a field and typical methods of commutative algebra and related algebraic geometry. This is intended in the sense of…
We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integro-differential…
Structures based on polarities have been used to provide relational semantics for propositional logics that are modelled algebraically by non-distributive lattices with additional operators. This article develops a first order notion of…
We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. We propose a general algebraic framework to find the solutions and to…