Related papers: Algebraic Geometry Over Four Rings and the Frontie…
We prove an upper bound for the number of rational points of bounded height on irreducible affine hypersurfaces. More precisely, given an irreducible polynomial $f \in \mathbb{Z}[X_1, \dots, X_n]$, we prove an upper bound on the number of…
Several problems in algebraic geometry and coding theory over finite rings are modeled by systems of algebraic equations. Among these problems, we have the rank decoding problem, which is used in the construction of public-key cryptography.…
We propose a novel algebraic framework for treating probability distributions represented by their cumulants such as the mean and covariance matrix. As an example, we consider the unsupervised learning problem of finding the subspace on…
Let V be a normal affine variety over the real numbers R, and let S be a semi-algebraic subset of V(R). We study the subring B(S) of the coordinate ring of V consisting of the polynomials that are bounded on S. We introduce the notion of…
We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining…
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 discuss existence of explicit search bounds for zeros of polynomials with coefficients in a number field. Our main result is a theorem about the existence of polynomial zeros of small height over the field of algebraic numbers outside of…
We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions…
In the problem of semialgebraic range searching, we are to preprocess a set of points in $\mathbb{R}^D$ such that the subset of points inside a semialgebraic region described by $O(1)$ polynomial inequalities of degree $\Delta$ can be found…
We study complexity measures on subsets of the boolean hypercube and exhibit connections between algebra (the Hilbert function) and combinatorics (VC theory). These connections yield results in both directions. Our main complexity-theoretic…
The earlier approach is used for description of qubits and geometric phase parameters, the things critical in the area of topological quantum computing. The used tool, Geometric (Clifford) Algebra is the most convenient formalism for that…
The Semialgebraic Orbit Problem is a fundamental reachability question that arises in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. An instance of the…
The field of numerical algebraic geometry consists of algorithms for numerically solving systems of polynomial equations. When the system is exact, such as having rational coefficients, the solution set is well-defined. However, for a…
We present a polynomial partitioning theorem for finite sets of points in the real locus of an irreducible complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean…
This survey article is concerned with the modeling of the kinematical structure of quantum systems in an algebraic framework which eliminates certain conceptual and computational difficulties of the conventional approaches. Relying on the…
Connectivity problems like k-Path and k-Disjoint Paths relate to many important milestones in parameterized complexity, namely the Graph Minors Project, color coding, and the recent development of techniques for obtaining kernelization…
This paper discusses the split feasibility problem with polynomials. The sets are semi-algebraic, defined by polynomial inequalities. They can be either convex or nonconvex, either feasible or infeasible. We give semidefinite relaxations…
In this paper we derive an upper bound for the degree of the strict invariant algebraic curve of a polynomial system in the complex project plane under generic condition. The results are obtained through the algebraic multiplicities of the…
These notes aim at providing a complete and systematic account of some foundational aspects of algebraic supergeometry, namely, the extension to the geometry of superschemes of many classical notions, techniques and results that make up the…
We sharpen and generalize the dimension growth bounds for the number of points of bounded height lying on an irreducible algebraic variety of degree $d$, over any global field. In particular, we focus on the affine hypersurface situation by…