Related papers: Shattered Sets and the Hilbert Function

Asking which sets are fixed-parameter tractable for a given parameterization constitutes much of the current research in parameterized complexity theory. This approach faces some of the core difficulties in complexity theory. By focussing…

The problem of computing spectra of operators is arguably one of the most investigated areas of computational mathematics. However, the problem of computing spectra of general bounded infinite matrices has only recently been solved. We…

For certain problems involving vector fields, it is possible to find an associated imaginary field that, in conjunction with the first, forms a complex field for which the equation can be solved. This result is generalized to arbitrary…

We consider the problem of realizing hyperbolicity cones as spectrahedra, i.e. as linear slices of cones of positive semidefinite matrices. The generalized Lax conjecture states that this is always possible. We use generalized Clifford…

We investigate systems of equations, involving parameters from the point of view of both control theory and computer algebra. The equations might involve linear operators such as partial (q-)differentiation, (q-)shift, (q-)difference as…

While non-contextual hidden-variable theories are proved to be impossible, contextual ones are possible. In a contextual hidden-variable theory, an observable is called a beable if the hidden-variable assigns its value in a given…

The tubal tensor framework provides a clean and effective algebraic setting for tensor computations, supporting matrix-mimetic features like Singular Value Decomposition and Eckart-Young-like optimality results. Underlying the tubal tensor…

Using Macaulay's correspondence we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for cactus varieties of the third Veronese embedding.…

Partial combinatory algebras are algebraic structures that serve as generalized models of computation. In this paper, we study embeddings of pcas. In particular, we systematize the embeddings between relativizations of Kleene's models, of…

In this work we present an extension of Chubanov's algorithm to the case of homogeneous feasibility problems over a symmetric cone K. As in Chubanov's method for linear feasibility problems, the algorithm consists of a basic procedure and a…

This work provides closed-form solutions and minimum achievable errors for a large class of low-rank approximation problems in Hilbert spaces. The proposed theorem generalizes to the case of bounded linear operators the previous results…

Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g. a graph is 3-colorable, hamiltonian, etc.) if and only if a related system…

A classic result of Lenstra [Math.~Oper.~Res.~1983] says that an integer linear program can be solved in fixed-parameter tractable (FPT) time for the parameter being the number of variables. We extend this result by incorporating…

The aim of this note is to exploit a new relationship between additive combinatorics and the geometry of monomial projective curves. We associate to a finite set of non-negative integers $A=\{a_1,\cdots, a_n\}$ a monomial projective curve…

We present a theory that produces several examples where the homotopy Lie algebra of a complex hyperplane arrangement is not finitely presented. We also present examples of hyperplane arrangements where the enveloping algebra of this Lie…

Consider the representations of an algebraic group G. In general, polynomial invariant functions may fail to separate orbits. The invariant subring may not be finitely generated, or the number and complexity of the generators may grow…

We study finite-dimensional representations of hyper loop algebras over non-algebraically closed fields. The main results concern the classification of the irreducible representations, the construction of the Weyl modules, base change,…

In this paper, we consider the quadratic programming problems under finitely many convex quadratic constraints in Hilbert spaces. By using the Legendre property of quadratic forms or the compactness of operators in the presentations of…

The H\"older continuity of the truncated moment map of a shade function in Euclidean space is established in the vicinity of a principal semi-algebraic set. The proof combines volume bounds of semi-algebraic sets and convex optimization…

While concepts and tools from Theoretical Computer Science are regularly applied to, and significantly support, software development for discrete problems, Numerical Engineering largely employs recipes and methods whose correctness and…