Related papers: Learning Algebraic Varieties from Samples
An uncomplicated and easily handling prescription that converts the task of checking the unitarity of massive, topologically massive, models into a straightforward algebraic exercise, is developed. The algorithm is used to test the…
We investigate stratified-algebraic vector bundles on a real algebraic variety X. A stratification of X is a finite collection of pairwise disjoint, Zariski locally closed subvarieties whose union is X. A topological vector bundle on X is…
In this article algebraic constructions are introduced in order to study the variety defined by a radical parametrization (a tuple of functions involving complex numbers, $n$ variables, the four field operations and radical extractions). We…
We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the $A$-discriminant in terms of Euler obstructions. Our primary goal is the development of reliable…
We discuss methods for using the Weil polynomial of an isogeny class of abelian varieties over a finite field to determine properties of the curves (if any) whose Jacobians lie in the isogeny class. Some methods are strong enough to show…
The aim of this paper is to study the behavior of Hodge-theoretic (intersection homology) genera and their associated characteristic classes under proper morphisms of complex algebraic varieties. We obtain formulae that relate (parametrized…
We show that the virtual Euler characteristics of the moduli spaces of $s$-pointed algebraic curves of genus $g$ can be determined from a polynomial in $1/\gamma$ where $\gamma$ permits specialization, through $\gamma=1,$ to the complex…
We study dynamical systems arising from word maps on simple groups. We develop a geometric method based on the classical trace map for investigating periodic points of such systems. These results lead to a new approach to the search of…
In this paper we explore by means of the method of Lagrangian descriptors the Julia sets arising from complex maps, and we analyze their underlying dynamics. In particular, we take a look at two classical examples: the quadratic mapping…
We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients.…
The aim of the paper is to discuss the relations between the three kinds of objects named in the title. In a sense, this is a survey of such relations; however, some new directions are also considered. This relates, especially, to sections…
These notes survey some basic results in toric varieties over a field with examples and applications. A computer algebra package (written by the second author) is described which deals with both affine and projective toric varieties in any…
We give an efficient, deterministic algorithm to decide if two abelian varieties over a number field are isogenous. From this, we derive an algorithm to compute the endomorphism ring of an elliptic curve over a number field.
This paper investigates the Jordan--Kronecker invariant of finite dimensional complex Lie algebras. We present an explicit algorithm for determining the type of a given Lie algebra from its Jordan--Kronecker invariant. The algorithm is…
The subject of persistent homology has vitalized applications of algebraic topology to point cloud data and to application fields far outside the realm of pure mathematics. The area has seen several fundamentally important results that are…
A strong link between information geometry and algebraic statistics is made by investigating statistical manifolds which are algebraic varieties. In particular it it shown how first and second order efficient estimators can be constructed,…
I will discuss recent progress by many people in the program of extending natural topological invariants from manifolds to singular spaces. Intersection homology theory and mixed Hodge theory are model examples of such invariants. The past…
We present a new algorithm to decide isomorphism between finite graded algebras. For a broad class of nilpotent Lie algebras, we demonstrate that it runs in time polynomial in the order of the input algebras. We introduce heuristics that…
Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, familiar from toric varieties and their moment maps. Among them are varieties of inverses…
We describe all polarizations for all abelian varieties over a finite field in a fixed isogeny class corresponding to a squarefree Weil polynomial, when one variety in the isogeny class admits a canonical liftings to characteristic zero,…