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We consider polynomially and rationally parameterized curves, where the polynomials in the parameterization have fixed supports and generic coefficients. We apply sparse (or toric) elimination theory in order to determine the vertex…
Nonclassical symmetries and reductions of polynomial equations and systems of polynomial equations are considered. It is shown that specific polynomial equations having "hidden" symmetries can be reduced to classical symmetric systems of…
This work investigates the use of sparse polynomial interpolation as a model order reduction method for the incompressible Navier-Stokes equations. Numerical results are presented underscoring the validity of sparse polynomial…
We present two effective tools for computing the positive tropicalization of algebraic varieties. First, we outline conditions under which the initial ideal can be used to compute the positive tropicalization, offering a real analogue to…
Porous electrodes are widely used in electrochemical systems, where accurately determining electric potentials, particularly overpotentials, is essential for understanding electrode behavior. At the macroscopic scale, porous electrodes are…
This paper studies the saddle point problem of polynomials. We give an algorithm for computing saddle points. It is based on solving Lasserre's hierarchy of semidefinite relaxations. Under some genericity assumptions on defining…
We propose in this paper a polynomial representation of TU-games, fuzzy measures, capacities, and more generally set functions. Our representation needs a countably infinite set of players and the natural ordering of finite sets of…
We prove the first polynomial bound on the number of monotonic homotopy moves required to tighten a collection of closed curves on any compact orientable surface, where the number of crossings in the curve is not allowed to increase at any…
Given a lattice polygon, we study the moduli space of all tropical plane curves with that Newton polygon. We determine a formula for the dimension of this space in terms of combinatorial properties of that polygon. We prove that if this…
Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial…
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
An algorithm for numerically computing the exponential of a matrix is presented. We have derived a polynomial expansion of $e^x$ by computing it as an initial value problem using a symbolic programming language. This algorithm is shown to…
We give new algorithms based on the sum-of-squares method for tensor decomposition. Our results improve the best known running times from quasi-polynomial to polynomial for several problems, including decomposing random overcomplete…
We use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solution in boxes with the side…
In general dimension, there is no known total polynomial algorithm for either convex hull or vertex enumeration, i.e. an algorithm whose complexity depends polynomially on the input and output sizes. It is thus important to identify…
We consider the tropicalization of tangent lines to a complete intersection curve $X$ in $\mathbb{P}^n$. Under mild hypotheses, we describe a procedure for computing the tropicalization of the image of the Gauss map of $X$ in terms of the…
We geometrically prove that in a d-dimensional cube with edges of length n, the number of particular d-dimensional tetrahedrons are given by Eulerian numbers. These tetrahedrons tassellate the cube, In this way the sum of the cubes are the…
The Fr\'{e}chet mean is a fundamental notion of central tendency defined as a minimizer of a sum of squared distances in a general metric space. In this paper, we study Fr\'{e}chet means in tropical geometry -- a piecewise linear,…
We study the problem of computing the isolated regular solutions of a system \((f_1,\ldots,f_n)\) of \(n\) polynomial equations in \(n\) variables \((X_1, \dots, X_n)\) over a field of characteristic zero \(k\). We focus on systems with a…
There are many connections between the invariants of the different powers of an ideal. We investigate how to construct minimal resolutions for all powers at once using methods from algebraic and polyhedral topology with a focus on ideals…