Related papers: Quantitative Combinatorial Nullstellensatz

Let $L$ be a linear differential operator with constant coefficients of order $n$ and complex eigenvalues $\lambda_{0},...,\lambda_{n}$. Assume that the set $U_{n}$ of all solutions of the equation $Lf=0$ is closed under complex…

A proof that minimum uncertainty states of the simplest periodic quantum system exist in a state space that is represented by a Colombeau algebra of generalised functions but not in Hilbert space or in the space of Schwartz distributions is…

We formulate and prove a Gelfand-Levitan trace formula for general quantum graphs with arbitrary edge lengths and coupling conditions which cover all self-adjoint operators on quantum graphs, except for a set of measure zero. The formula is…

Nowhere dense classes of graphs are very general classes of uniformly sparse graphs with several seemingly unrelated characterisations. From an algorithmic perspective, a characterisation of these classes in terms of uniform quasi-wideness,…

In the first 36 pages of this paper, we provide polynomial quantum algorithms for additive approximations of the Tutte polynomial, at any point in the Tutte plane, for any planar graph. This includes as special cases the AJL algorithm for…

Classification theorems for linear differential equations in two real variables, possessing eigenfunctions in the form of the polynomials (the generalized Bochner problem) are given. The main result is based on the consideration of the…

We compute the cohomology ring of a generalised type of configuration space of points in $\mathbb{R}^r$. This configuration space is indexed by a graph. In the case the graph is complete the result is known and it is due to Arnold and…

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…

Motivated by applications in combinatorial geometry, we consider the following question: Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_m)$ be an $m$-partition of a positive integer $n$, $S_i \subseteq \mathbb{C}^{\lambda_i}$ be finite…

We prove a theorem on algebraic osculation and we apply our result to the Computer Algebra problem of polynomial factorization. We consider X a smooth completion of the complex plane and D an effective divisor supported on the boundary of…

We show that algebraic formulas and constant-depth circuits are closed under taking factors. In other words, we show that if a multivariate polynomial over a field of characteristic zero has a small constant-depth circuit or formula, then…

The "openness" of a complex polynomial mapping is discussed and applied to the Fundamental Theorem of Algebra. In this category fall proofs of S. Wolfenstein, R.L. Thompson, J. Milnor, and S. Reich-S. Smale. These proofs take into account…

It is known that a graph isomorphism testing algorithm is polynomially equivalent to a detecting of a graph non-trivial automorphism algorithm. The polynomiality of the latter algorithm, is obtained by consideration of symmetry properties…

Let $K$ be an arbitrary field of characteristic zero, $P_n:= K[ x_1, ..., x_n]$ be a polynomial algebra, and $P_{n, x_1}:= K[x_1^{-1}, x_1, ..., x_n]$, for $n\geq 2$. Let $\s' \in {\rm Aut}_K(P_n)$ be given by $$ x_1\mapsto x_1-1, \quad…

In this monograph, we lay the foundations for a new theory that generalizes real algebraic geometry. Let $R|K$ be a field extension, where $R$ is a real closed field and $K$ is an ordered subfield of $R$. The main objective is to study…

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…

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 consider matrices of the form $qD+A$, with $D$ being the diagonal matrix of degrees, $A$ being the adjacency matrix, and $q$ a fixed value. Given a graph $H$ and $B\subseteq V(G)$, which we call a coalescent pair $(H,B)$, we derive a…

Formulas for the solutions of initial value problems for ordinary differential equations with singular $\delta^{(n)}$-like driving terms are derived in the framework of an algebra of generalized functions (of Colombeau type) over a field of…

We show that if $P$ is a lattice polytope in the nonnegative orthant of $\R^k$ and $\chi$ is a coloring of the lattice points in the orthant such that the color $\chi(a+b)$ depends only on the colors $\chi(a)$ and $\chi(b)$, then the number…