Related papers: Numerical homotopies to compute generic points on …
We establish basic facts about the varieties of homogeneous polynomials divisible by powers of linear forms, and explain consequences for geometric complexity theory. This includes quadratic set-theoretic equations, a description of the…
Homotopy approaches to Bayesian inference have found widespread use especially if the Kullback-Leibler divergence between the prior and the posterior distribution is large. Here we extend one of these homotopy approach to include an…
Suppose $F:=(f_1,\ldots,f_n)$ is a system of random $n$-variate polynomials with $f_i$ having degree $\leq\!d_i$ and the coefficient of $x^{a_1}_1\cdots x^{a_n}_n$ in $f_i$ being an independent complex Gaussian of mean $0$ and variance…
We present a general method for constructing real solutions to some problems in enumerative geometry which gives lower bounds on the maximum number of real solutions. We apply this method to show that two new classes of enumerative…
We present fully polynomial approximation schemes for a broad class of Holant problems with complex edge weights, which we call Holant polynomials. We transform these problems into partition functions of abstract combinatorial structures…
In this work we introduce and analyze a new multiscale method for strongly nonlinear monotone equations in the spirit of the Localized Orthogonal Decomposition. A problem-adapted multiscale space is constructed by solving linear local…
The requirement for solving a polynomial is a means of breaking its symmetry, which in the case of the quintic, is that of the symmetric group S_5. Induced by its five-dimensional linear permutation representation is a three-dimensional…
A simple algorithm to compute all the zeros of a generic polynomial is proposed.
We consider positive solutions to parametrized systems of generalized polynomial equations (with real exponents and positive parameters). By a fundamental result obtained in parallel work, polynomial systems are determined by geometric…
A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness…
Polytope complexes are the generalisation of polygon meshes in geo-information systems (GIS) to arbitrary dimension, and a natural concept for accessing spatio-temporal information. Complexes of each dimension have a straight-forward…
We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades.…
This article develops a new predictor-corrector algorithm for numerical path tracking in the context of polynomial homotopy continuation. In the corrector step it uses a newly developed Newton corrector algorithm which rejects an initial…
This paper presents a noncommutative theory of symmetric functions, based on the notion of quasi-determinant. We begin with a formal theory, corresponding to the case of symmetric functions in an infinite number of independent variables.…
In this article we give a computational study of combinatorics of the discriminantal arrangements. The discriminantal arrangements are parametrized by two positive integers n and k such that n>k. The intersection lattice of the…
Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed. A general method of construction of…
A theory of numerical path-following in toric varieties was suggested in two previous papers. The motivation is solving systems of polynomials with real or complex coefficients. When those polynomials are not assumed 'dense', solving them…
The main aim of this paper is to provide a method which allows finding limiting shapes of symbolic generic initial systems of higher-dimensional subvarieties of P^n. M. Mustata and S. Mayes established a connection between volumes of…
A polynomial homotopy is a family of polynomial systems, where the systems in the family depend on one parameter. If for one value of the parameter we know a regular solution, then what is the nearest value of the parameter for which the…
We construct an effective algorithmic method to compute the homological monodromy of a complex polynomial which is tame. As an application we show the existence of conjugated polynomials in a number field which are not topologically…