Related papers: Motivated exposition of combinatorial Nullstellens…
We prove a general version of Bezout's form of the Nullstellensatz for arbitrary fields. The corresponding sufficient and necessary condition only involves the local existence of multi-valued roots for each of the polynomials belonging to…
Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of…
It is shown that by eliminating duality theory of vector spaces from a recent proof of Kouba (O. Kouba, A duality based proof of the Combinatorial Nullstellensatz. Electron. J. Combin. 16 (2009), #N9) one obtains a direct proof of the…
This article gives a class of Nullstellens\"atze for noncommutative polynomials. The singularity set of a noncommutative polynomial $f=f(x_1,\dots,x_g)$ is $Z(f)=(Z_n(f))_n$, where $Z_n(f)=\{X \in M_n^g: \det f(X) = 0\}.$ The first main…
This is a detailed survey -- with rigorous and self-contained proofs -- of some of the basics of elementary combinatorics and algebra, including the properties of finite sums, binomial coefficients, permutations and determinants. It is…
We prove a Nullstellensatz for the ring of polynomial functions in n non-commuting variables over Hamilton's ring of real quaternions. We also characterize the generalized polynomial identities in n variables which hold over the…
In this paper we outline a Matrix Ansatz approach to some problems of combinatorial enumeration. The idea is that many interesting quantities can be expressed in terms of products of matrices, where the matrices obey certain relations. We…
We introduce a new form of the polynomial method based on what we call "shift operators," which we use to give efficient and intuitive new proofs of results previously shown using a wide range of polynomial methods, including Alon's…
The aim of this note is to present an easy proof of Hilbert's Nullstellensatz using Groebner basis. I believe, that the proof has some methodical advantage in a course on Groebner bases. Key words: Hilbert's Nullstellensatz, Groebner bases.
We consider the numbers arising in the problem of normal ordering of expressions in canonical boson creation and annihilation operators. We treat a general form of a boson string which is shown to be associated with generalizations of…
Hilbert's Nullstellensatz is one of the most fundamental correspondences between algebra and geometry, and has inspired a plethora of noncommutative analogs. In last two decades, there has been an increased interest in understanding…
In this article, we use the Combinatorial Nullstellensatz to give new proofs of the Cauchy-Davenport, the Dias da Silva-Hamidoune and to generalize a previous addition theorem of the author. Precisely, this last result proves that for a set…
This article extends the classical Real Nullstellensatz to matrices of polynomials in a free $\ast$-algebra $\RR\axs$ with $x=(x_1, \ldots, x_n)$. This result is a generalization of a result of Cimpri\vc, Helton, McCullough, and the author.…
We describe a solution of the word problem in free fields (coming from non-commutative polynomials over a commutative field) using elementary linear algebra, provided that the elements are given by minimal linear representations. It relies…
The purpose of this note is to attach a name to a natural class of combinatorial problems and to point out that this class includes many important special cases. We also show that a simple problem of placing nonoverlapping labels on a…
This is an expository introduction to simplicial sets and simplicial homotopy theory with particular focus on relating the combinatorial aspects of the theory to their geometric/topological origins. It is intended to be accessible to…
One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some…
Motivated by higher vanishing multiplicity generalizations of Alon's Combinatorial Nullstellensatz and its applications, we study the following problem: for fixed $k\geq 1$ and $n$ large with respect to $k$, what is the minimum possible…
We are often interested in decomposing complex, structured data into simple components that explain the data. The linear version of this problem is well-studied as dictionary learning and factor analysis. In this work, we propose a…
We prove a version of a Nullstellensatz for partial exponential fields $(K,E)$, even though the ring of exponential polynomials $K[X_1,\ldots,X_n]^E$ is not a Hilbert ring. We show that under certain natural conditions one can embed an…