Related papers: Enumerative Aspects of Nullstellensatz Certificate…

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…

We introduce a low-memory framework for certifying numerical solutions to polynomial systems which uses solution iterators and spatial partitioning trees to reduce memory requirements. We provide a prototypical algorithm, analyze its…

Schubert polynomials are a basis for the polynomial ring that represent Schubert classes for the flag manifold. In this paper, we introduce and develop several new combinatorial models for Schubert polynomials that relate them to other…

A non-negativity certificate (NNC) is a way to write a polynomial so that its non-negativity on a semialgebraic set becomes evident. Positivstellens\"atze (Ps\"atze) guarantee the existence of NNCs. Both, NNCs and Ps\"atze underlie powerful…

In recent years, techniques based on convex optimization and real algebra that produce converging hierarchies of lower bounds for polynomial minimization problems have gained much popularity. At their heart, these hierarchies rely crucially…

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…

We compile a long list of equivalent formulations of Hilbert's Nullstellensatz in infinite dimensions, and prove a persistence result for the strong Nullstellensatz in large polynomial rings.

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 present new generalizations of Olson's theorem and of a consequence of Alon's Combinatorial Nullstellensatz. These enable us to extend some of their combinatorial applications with conditions modulo primes to conditions modulo prime…

In this article we combine two developments in polynomial optimization. On the one hand, we consider nonnegativity certificates based on sums of nonnegative circuit polynomials, which were recently introduced by the second and the third…

Binary quadratic Diophantine equations are of interest from the viewpoint of computational complexity theory. They contain as special cases many examples of natural problems apparantly occupying intermediate stages in the P-NP hierarchy,…

Applying Gr\"obner basis theory to concrete problems in Lean 4 remains difficult since the current formalization of multivariate polynomials is based on a non-computable representation and is therefore not suitable for efficient symbolic…

The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, yields insight into the hierarchy of complexity classes. In particular, the theory produces algebraic tests for a…

Many practical problems in almost all scientific and technological disciplines have been classified as computationally hard (NP-hard or even NP-complete). In life sciences, combinatorial optimization problems frequently arise in molecular…

We discuss the use of methods coming from integrable systems to study problems of enumerative and algebraic combinatorics, and develop two examples: the enumeration of Alternating Sign Matrices and related combinatorial objects, and the…

We consider the problem of enumeration of planar maps and revisit its one-matrix model solution in the light of recent combinatorial techniques involving conjugated trees. We adapt and generalize these techniques so as to give an…

A new symbolic algorithm to compute sums of squares multipliers (certificates) to witness the membership of non-negative univariate polynomials in a saturated univariate quadratic module is presented. Certificates are first computed in…

Hilbert's Nullstellensatz is a fundamental result in algebraic geometry that gives a necessary and sufficient condition for a finite collection of multivariate polynomials to have a common zero in an algebraically closed field. Associated…

Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry…

Amoebas and coamoebas are the logarithmic images of algebraic varieties and the images of algebraic varieties under the arg-map, respectively. We present new techniques for computational problems on amoebas and coamoebas, thus establishing…