Related papers: Real Algebraic Geometry for Geometric Constraints
The notion of hidden symmetry algebra used in the context of exactly solvable systems is re-examined from the purely algebraic way, analyzing subspaces of commuting polynomials that generate finite-dimensional quadratic algebras. By…
The observational characteristics of a linear structural equation model can be effectively described by polynomial constraints on the observed covariance matrix. However, these polynomials can be exponentially large, making them impractical…
Much of the success of deep learning is drawn from building architectures that properly respect underlying symmetry and structure in the data on which they operate - a set of considerations that have been united under the banner of…
This book is expository and is in Russian. It is shown how in the course of solution of interesting geometric problems (close to applications) naturally appear main notions of algebraic topology (homology groups, obstructions and…
The parametric geometry of numbers has allowed to visualize the simultaneous approximation properties of a collection of real numbers through the combined graph of the related successive minima functions. Several inequalities among…
Symmetries and reductions of some algebraic equations are considered. Transformations that preserve the form of several algebraic equations, as well as transformations that reduce the degree of these equations, are described. Illustrative…
We aim at studying collections of algebraic structures defined over a commutative ring and investigating the complexity of significant constructions carried out on these objects. The assignment of measures of size, via a multiplicity…
Geometry problem solving, a crucial aspect of mathematical reasoning, is vital across various domains, including education, the assessment of AI's mathematical abilities, and multimodal capability evaluation. The recent surge in deep…
We introduce the notion of a robust parameterized arithmetic circuit for the evaluation of algebraic families of multivariate polynomials. Based on this notion, we present a computation model, adapted to Scientific Computing, which captures…
Linear Geometry studies geometric properties which can be expressed via the notion of a line. All information about lines is encoded in a ternary relation called a line relation. A set endowed with a line relation is called a liner. So,…
We use algebraic geometry to study matrix rigidity, and more generally, the complexity of computing a matrix-vector product, continuing a study initiated by Kumar, et. al. We (i) exhibit many non-obvious equations testing for (border)…
Numerical data structures for positive dimensional solution sets of polynomial systems are sets of generic points cut out by random planes of complimentary dimension. We may represent the linear spaces defined by those planes either by…
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…
Geometry is a fundamental part of robotics and there have been various frameworks of representation over the years. Recently, geometric algebra has gained attention for its property of unifying many of those previous ideas into one algebra.…
We survey both old and new developments in the theory of algorithms in real algebraic geometry -- starting from effective quantifier elimination in the first order theory of reals due to Tarski and Seidenberg, to more recent algorithms for…
The exact complexity of geometric cuts and bisections is the longstanding open problem including even the dimension one. In this paper, we resolve this problem for dimension one (the real line) by designing an exact polynomial time…
The canonical projections of the unit spheres are generalized to special generic maps and round fold maps, for example. They are generalizations from the viewpoint of singularity theory of differentiable maps and these maps restrict the…
We discuss new problems in universal algebraic geometry and explain them by boolean equations.
We develop a new framework of relative algebroids to address existence and classification problems of geometric structures subject to partial differential equations.
Kinematic algebras can be realised on geometric spaces and constrain the physical models that can live on these spaces. Different types of kinematic algebras exist and we consider the interplay of these algebras for non-relativistic limits…