Related papers: Algorithmic Semi-algebraic Geometry and Topology -…
In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given. From this result, here we derive applications based on level curves to determine some…
We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for {\em graphs with semi-edges}. The notion of graph covering is a discretization of coverings between surfaces or topological…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
We develop the idea of using an algebraic-geometry approach to classical differential geometry problems. Consider an orthogonal net constructed according to algebraic-geometric data we obtain a set of smooth orthogonal nets that are…
This work presents a new evolutionary optimization algorithm in theoretical mathematics with important applications in scientific computing. The use of the evolutionary algorithm is justified by the difficulty of the study of the…
This paper is an introductory and informal exposition on the topology of polygonal meshes. We begin with a broad overview of topological notions and discuss how homeomorphisms, homotopy, and homology can be used to characterise topology. We…
The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?" The question has been asked before but as yet the leading algorithms for answering such…
Recent results in control systems and numerical integration literature utilize invariant set theory to lift dynamical systems evolving on nonlinear manifolds to those evolving on vector spaces. We leverage this technique to propose an…
The problem of optimizing over random structures emerges in many areas of science and engineering, ranging from statistical physics to machine learning and artificial intelligence. For many such structures finding optimal solutions by means…
In this paper we introduce the concept of characteristic number that are proven to be useful in the study of the combinatorics of graph cohomology. We claim that it is a good combinatorial counterpart for geometric Betti numbers. We then…
In this paper, we present a deterministic algorithm to find a strong generic position for an algebraic space curve. We modify our existing algorithm for computing the topology of an algebraic space curve and analyze the bit complexity of…
Enumerative Geometry is concerned with the number of solutions to a structured system of polynomial equations, when the structure comes from geometry. Enumerative real algebraic geometry studies real solutions to such systems, particularly…
This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical…
We characterize the canonical algebras such that for all dimension vectors of homogeneous modules the corresponding module varieties are complete intersections (respectively, normal). We also investigate the sets of common zeros of…
This paper provides an overview of the main ideas driving the bootstrap algebraic multigrid methodology, including compatible relaxation and algebraic distances for defining effective coarsening strategies, the least squares method for…
We give new results for problems in computational and statistical machine learning using tools from high-dimensional geometry and probability. We break up our treatment into two parts. In Part I, we focus on computational considerations in…
We report on implementations for algorithms treating algebraic and arithmetic properties of hypergeometric functions in the computer algebra system SageMath. We treat hypergeometric series over the rational numbers, over finite fields, and…
We consider how the problem of determining normal forms for a specific class of nonholonomic systems leads to various interesting and concrete bridges between two apparently unrelated themes. Various ideas that traditionally pertain to the…
This paper focuses on polynomial dynamical systems over finite fields. These systems appear in a variety of contexts, in computer science, engineering, and computational biology, for instance as models of intracellular biochemical networks.…
We show that every finite semilattice can be represented as an atomized semilattice, an algebraic structure with additional elements (atoms) that extend the semilattice's partial order. Each atom maps to one subdirectly irreducible…