Related papers: Degree Bounds for Polynomial Verification of the M…
We propose a sufficient condition for invertibility of a polynomial mapping function defined on a cube or simplex. This condition is applicable to finite element analysis using curved meshes. The sufficient condition is based on an analysis…
It is shown that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has positive coefficients when $m = 6$ and $A$ and $B$ are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior,…
We consider the stability analysis of a large class of linear 1-D PDEs with polynomial data. This class of PDEs contains, as examples, parabolic and hyperbolic PDEs, PDEs with boundary feedback and systems of in-domain/boundary coupled…
For a given computational problem, a certificate is a piece of data that one (the prover) attaches to the output with the aim of allowing efficient verification (by the verifier) that this output is correct. Here, we consider the minimal…
We consider two algorithms which can be used for proving positivity of sequences that are defined by a linear recurrence equation with polynomial coefficients (P-finite sequences). Both algorithms have in common that while they do succeed…
We investigate whether the group algebra of a finite group over a localisation of the integers is semiperfect. The main result is a necessary and sufficient arithmetic criterion in the ordinary case. In the modular case, we propose a…
We consider linear recurrences with polynomial coefficients of Poincar\'e type and with a unique simple dominant eigenvalue. We give an algorithm that proves or disproves positivity of solutions provided the initial conditions satisfy a…
We consider the problem of approximating the reachable set of a discrete-time polynomial system from a semialgebraic set of initial conditions under general semialgebraic set constraints. Assuming inclusion in a given simple set like a box…
This paper establishes new upper bounds for the right eigenvalues of monic matrix polynomials over the quaternion division algebra. The noncommutative nature of quaternion multiplication presents fundamental challenges in eigenvalue…
Certificates to a linear algebra computation are additional data structures for each output, which can be used by a---possibly randomized---verification algorithm that proves the correctness of each output. The certificates are essentially…
The row (column) completion problem of polynomial matrices of given degree with prescribed eigenstructure has been studied in \cite{AmBaMaRo23}, where several results of prescription of some of the four types of invariants that form the…
Given an $\mathcal{H}$-polytope $P$ and a $\mathcal{V}$-polytope $Q$, the decision problem whether $P$ is contained in $Q$ is co-NP-complete. This hardness remains if $P$ is restricted to be a standard cube and $Q$ is restricted to be the…
We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the positive…
One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some…
Separability criteria are typically of the necessary, but not sufficient, variety, in that satisfying some separability criterion, such as positivity of eigenvalues under partial transpose, does not strictly imply separability. Certifying…
Current techniques for generating a knowledge space, such as QUERY, guarantees that the resulting structure is closed under union, but not that it satisfies wellgradedness, which is one of the defining conditions for a learning space. We…
The singularity degree of a semidefinite programming problem is the smallest number of facial reduction steps to make the problem strictly feasible. We introduce two new graph parameters, called the singularity degree and the nondegenerate…
The question how to certify non-negativity of a polynomial function lies at the heart of Real Algebra and also has important applications to Optimization. In this article we investigate the question of non-negativity in the context of…
As a partial answer to a question of Rao, a deterministic and customizable efficient algorithm is presented to test whether an arbitrary graphical degree sequence has a bipartite realization. The algorithm can be configured to run in…
We investigate the power of graph isomorphism algorithms based on algebraic reasoning techniques like Gr\"obner basis computation. The idea of these algorithms is to encode two graphs into a system of equations that are satisfiable if and…