Related papers: Learning Algebraic Varieties from Samples
Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional…
This is a survey on the use of low-degree polynomials to predict and explain the apparent statistical-computational tradeoffs in a variety of average-case computational problems. In a nutshell, this framework measures the complexity of a…
We address the question of identifying non-smooth points in affine real algebraic varieties. A simple algebraic criterion will be formulated and proven. As an application we can answer several questions about the configuration spaces of…
A very first step to develop non-commutative algebraic geometry is the arithmetic of polynomials in non-commuting variables over a commutative field, that is, the study of elements in free associative algebras. This investigation is…
We are concerned with the problem of decomposing the parameter space of a parametric system of polynomial equations, and possibly some polynomial inequality constraints, with respect to the number of real solutions that the system attains.…
We explore to what extent the underlying variety of a connected algebraic group or the underlying manifold of a real Lie group determines its group structure.
In this paper, we revisit the problem of classifying real algebraic and semialgebraic sets by their topological types, focusing on establishing the effectiveness of bounds rather than deriving new quantitative estimates. Building on Hardt's…
In this paper, we consider the simplest class of stratified spaces -- linearly embedded graphs. We present an algorithm that learns the abstract structure of an embedded graph and models the specific embedding from a point cloud sampled…
Laminations are a combinatorial and topological way to study Julia sets. Laminations give information about the structure of parameter space of degree $d$ polynomials with connected Julia sets. We first study fixed point portraits in…
Let $A$ be an abelian variety with commutative endomorphism algebra over a finite field $k$. The $k$-isogeny class of $A$ is uniquely determined by a Weil polynomial $f_A$ without multiple roots. We give a classification of the groups of…
Primarily this paper presents an expository report on alternatives to the traditional methods of classifying representations of finite dimensional algebras. Some new results illustrating such alternatives for algebras with only finitely…
We prove the existence of infinitely many solutions to an elliptic problem by borrowing the techniques from algebraic topology. The solution(s) thus obtained will also be proved to be bounded.
We give upper and lower bounds on the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian…
The concept of viewing graph solvability has gained significant interest in the context of structure-from-motion. A viewing graph is a mathematical structure where nodes are associated to cameras and edges represent the epipolar geometry…
We suggest to look at formal sentences describing complex algebraic varieties together with their universal covers as topological invariants. We prove that for abelian varieties and Shimura varieties this is indeed a complete invariant,…
The survey gives an overview of the achievements in topology of real algebraic varieties in the direction initiated in the early 70th by V.I.Arnold and V.A.Rokhlin. We make an attempt to systematize the principal results in the subject.…
Computation of polynomial relative invariants is a classical tool in algebra. Relative differential invariants are central for the equivalence problem of geometric structures. We address the fundamental problem of finite generation of their…
Profinite algebras are exactly those that are isomorphic to inverse limits of finite algebras. Such algebras are naturally equipped with Boolean topologies. A variety $\mathcal V$ is standard if every Boolean topological algebra with the…
Assume that a finite set of points is randomly sampled from a subspace of a metric space. Recent advances in computational topology have provided several approaches to recovering the geometric and topological properties of the underlying…
The distortion varieties of a given projective variety are parametrized by duplicating coordinates and multiplying them with monomials. We study their degrees and defining equations. Exact formulas are obtained for the case of one-parameter…