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We develop a collection of numerical algorithms which connect ideas from polyhedral geometry and algebraic geometry. The first algorithm we develop functions as a numerical oracle for the Newton polytope of a hypersurface and is based on…
A theorem of Kushnirenko and Bernstein shows that the number of isolated roots of a system of polynomials in a torus is bounded above by the mixed volume of the Newton polytopes of the given polynomials, and this upper bound is generically…
In this paper we develop in detail the geometric constructions that lead to many uniqueness results for the determination of polyhedral sets, typically scatterers, by a finite minimal number of measurements. We highlight how unique…
We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for…
The nonvanishing problem asks if a coefficient of a polynomial is nonzero. Many families of polynomials in algebraic combinatorics admit combinatorial counting rules and simultaneously enjoy having saturated Newton polytopes (SNP). Thereby,…
We study polyhedral approximations to the cone of nonnegative polynomials. We show that any constant ratio polyhedral approximation to the cone of nonnegative degree $2d$ forms in $n$ variables has to have exponentially many facets in terms…
A system of singular integral equations with monotone and concave nonlinearity in the subcritical case is investigated. The specified system and its scalar analog have direct applications in various areas of physics and biology. In…
The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this…
Solutions to many important partial differential equations satisfy bounds constraints, but approximations computed by finite element or finite difference methods typically fail to respect the same conditions. Chang and Nakshatrala enforce…
The holy grail of deep learning is to come up with an automatic method to design optimal architectures for different applications. In other words, how can we effectively dimension and organize neurons along the network layers based on the…
We give a short survey on computational techniques which can be used to solve the representation conversion problem for polyhedra up to symmetries. We in particular discuss decomposition methods, which reduce the problem to a number of…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
Polyhedral estimate is a generic efficiently computable nonlinear in observations routine for recovering unknown signal belonging to a given convex compact set from noisy observation of signal's linear image. Risk analysis and optimal…
We show that Hilbert's Nullstellensatz, the problem of deciding if a system of multivariate polynomial equations has a solution in the algebraic closure of the underlying field, lies in the counting hierarchy. More generally, we show that…
Covering numbers are a powerful tool used in the development of approximation algorithms, randomized dimension reduction methods, smoothed complexity analysis, and others. In this paper we prove upper bounds on the covering number of…
We study a graded vector space of polynomials associated to a square matrix, defined by a finite difference condition along the rows. We show this space coincides with one defined by directional derivatives, and prove it is…
Given a pure, full-dimensional, locally strongly connected polyhedral complex C with convex support, we characterize, by a local codimension-2 condition, polyhedral complexes that coarsen C. The proof of the characterization draws upon a…
This paper studies the complexity of matrix Putinar's Positivstellens{\"a}tz on the semialgebraic set that is given by the polynomial matrix inequality. \rev{When the quadratic module generated by the constrained polynomial matrix is…
In this article we review some problems in physics, chemistry and mathematics that lead naturally to a class of polyhedra which include the Platonic solids. Examples include the study of electrons on a sphere, cages of carbon atoms, central…
Based on previous work by the author we deduce that the invariant introduced by Bierstone and Milman in order to give a proof for constructive resolution of singularities in characteristic zero can be determined purely by considering…