Related papers: VPSPACE and a Transfer Theorem over the Reals

We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the…

The computational complexity of polynomial ideals and Gr\"obner bases has been studied since the 1980s. In recent years, the related notions of polynomial subalgebras and SAGBI bases have gained more and more attention in computational…

In this paper, we study polynomial-like elements in vector spaces equipped with group actions. We first define these elements via iterated difference operators. In the case of a full rank lattice acting on an Euclidean space, these…

We study the verification of distributed systems where processes are finite automata with access to a shared pool of locks. We consider objectives that are boolean combinations of local regular constraints. We show that the problem,…

We study homomorphism polynomials, which are polynomials that enumerate all homomorphisms from a pattern graph $H$ to $n$-vertex graphs. These polynomials have received a lot of attention recently for their crucial role in several new…

The aim of this paper is twofold. The first is to give a quantitative version of Schmidt's subspace theorem for arbitrary families of higher degree polynomials. The second is to give a generalization of the subspace theorem for arbitrary…

Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares…

In this paper we give an Immerman's Theorem for real-valued computation. We define circuits operating over real numbers and show that families of such circuits of polynomial size and constant depth decide exactly those sets of vectors of…

We consider the numerical irreducible decomposition of a positive dimensional solution set of a polynomial system into irreducible factors. Path tracking techniques computing loops around singularities connect points on the same irreducible…

This paper discusses the split feasibility problem with polynomials. The sets are semi-algebraic, defined by polynomial inequalities. They can be either convex or nonconvex, either feasible or infeasible. We give semidefinite relaxations…

We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\Pi_{2^{m}}^*$ represent the set of polynomials computable with…

We consider the problem of determining which classes of functions can be tested more efficiently than they can be learned, in the distribution-free sample-based model that corresponds to the standard PAC learning setting. Our main result…

We study how the adoption of an evaluation mechanism with sharing and memoization impacts the class of functions which can be computed in polynomial time. We first show how a natural cost model in which lookup for an already computed value…

Abstract notions of convexity over the vertices of a graph, and corresponding notions of halfspaces, have recently gained attention from the machine learning community. In this work we study monophonic halfspaces, a notion of graph…

In this paper, we generalize an elementary real-analysis result to a class of topological vector spaces. We also give an example of a topological vector space to which the result cannot be generalized.

We consider several families of combinatorial polytopes associated with the following NP-complete problems: maximum cut, Boolean quadratic programming, quadratic linear ordering, quadratic assignment, set partition, set packing, stable set,…

A Pfaffian circuit is a tensor contraction network where the edges are labeled with changes of bases in such a way that a very specific set of combinatorial properties are satisfied. By modeling the permissible changes of bases as systems…

Matrix polynomials given in an orthogonal basis are considered. Following the ideas of Mackey et al. "Vector spaces of Linearizations for Matrix Polynomials" (2006), the vec- tor spaces, called M1(P), M2(P) and DM(P), of potential…

A new class of distributional transformations is introduced, characterized by equations relating function weighted expectations of test functions on a given distribution to expectations of the transformed distribution on the test function's…

Fewnomial theory began with explicit bounds -- solely in terms of the number of variables and monomial terms -- on the number of real roots of systems of polynomial equations. Here we take the next logical step of investigating the…