Related papers: Polyhedral Homotopies in Cox Coordinates

Polynomial system solving has seen major progress in both theory and practice over the past decade. A landmark achievement was addressing Smale's 17th problem, establishing average-case polynomial-time algorithms for computing approximate…

The goal of Point Distance Solving Problems is to find 2D or 3D placements of points knowing distances between some pairs of points. The common guideline is to solve them by a numerical iterative method (\emph{e.g.} Newton-Raphson method).…

We introduce a cohomological method to compute Cox rings of hypersurfaces in the ambient space P^1 x P^n, which is more direct than existing methods. We prove that smooth hypersurfaces defined by regular sequences of coefficients are Mori…

Finding the solutions to a system of multivariate polynomial equations is a fundamental problem in mathematics and computer science. It involves evaluating the polynomials at many points, often chosen from a grid. In most current methods,…

Parametric search has been widely used in geometric algorithms. Cole's improvement provides a way of saving a logarithmic factor in the running time over what is achievable using the standard method. Unfortunately, this improvement comes at…

Let X^{circ} be the space of all labeled tetrahedra in P^{3}. In [BGS] we constructed a smooth symmetric compactification X-tilde of X^{circ}. In this article we show that the complement X-tilde smallsetminus X^{circ} is a divisor with…

In this article we consider the homotopy theory of stratified spaces through a simplicial point of view. We first consider a model category of filtered simplicial sets over some fixed poset $P$, and show that it is a simplicial…

The singular simplicial set Sing(X) of a space X completely captures its weak homotopy type. We introduce a category of_controlled sets_, yielding _simplicial controlled sets_, such that one can functorially produce a singular simplicial…

We present a new algorithm for solving the real roots of a bivariate polynomial system $\Sigma=\{f(x,y),g(x,y)\}$ with a finite number of solutions by using a zero-matching method. The method is based on a lower bound for bivariate…

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…

We study Cox rings of normal threefolds on which SL2 acts with a dense orbit. Exploiting the method of U-invariants, we obtain combinatorial criteria for the total coordinate space and the base variety to have log terminal singularities.…

This paper proposes a real moment-HSOS hierarchy for complex polynomial optimization problems with real coefficients. We show that this hierarchy provides the same sequence of lower bounds as the complex analogue, yet is much cheaper to…

An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the six-order boussinesq equation. We summarize the general formulas for similarity reduction solutions and similarity reduction equations of…

This paper studies the copositive optimization problem whose objective is a sparse polynomial, with linear constraints over the nonnegative orthant. We propose sparse Moment-SOS relaxations to solve it. Necessary and sufficient conditions…

To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible…

A presheaf of complexes is constructed on a category of weighted finite subsets of a fixed Euclidean space. To each object, a Koszul complex is assigned which resolves the coordinate ring of least squares solutions on that data set for a…

In this article we consider the application of Euler's homogeneous function theorem together with Stokes' theorem to exactly integrate families of polynomial spaces over general polygonal and polyhedral (polytopic) domains in two- and…

We exhibit a probabilistic symbolic algorithm for solving zero-dimensional sparse systems. Our algorithm combines a symbolic homotopy procedure, based on a flat deformation of a certain morphism of affine varieties, with the polyhedral…

The hamiltonian circuit polytope is the convex hull of feasible solutions for the circuit constraint, which provides a succinct formulation of the traveling salesman and other sequencing problems. We study the polytope by establishing its…

We present a new continuation algorithm to find all nondegenerate real solutions to a system of polynomial equations. Unlike homotopy methods, it is not based on a deformation of the system; instead, it traces real curves connecting the…