Related papers: Computing Certificates of Strictly Positive Polyno…
A novel method with two variations is proposed with which the number of positive and negative zeros of a polynomial with real coefficients and degree $n$ can be restricted with significantly better determinacy than that provided by the…
Motivated by Wilmshurst's conjecture, we investigate the zeros of harmonic polynomials. We utilize a certified counting approach which is a combination of two methods from numerical algebraic geometry: numerical polynomial homotopy…
Current methods for verifying quantum computers are predominately based on interactive or automatic theorem provers. Considering that quantum computers are dynamical in nature, this paper employs and extends the concepts from the…
We propose a novel method for computing exact pointwise robustness of deep neural networks for all convex $\ell_p$ norms. Our algorithm, GeoCert, finds the largest $\ell_p$ ball centered at an input point $x_0$, within which the output…
We propose new supermartingale-based certificates for verifying almost sure satisfaction of $\omega$-regular properties: (1) generalised Streett supermartingales (GSSMs) and their lexicographic extension (LexGSSMs), (2) distribution-valued…
We show that existence of a global polynomial Lyapunov function for a homogeneous polynomial vector field or a planar polynomial vector field (under a mild condition) implies existence of a polynomial Lyapunov function that is a sum of…
In this paper we consider the problem of how to computationally test whether a matrix inequality is positive semidefinite on a semialgebraic set. We propose a family of sufficient conditions using the theory of matrix Positivstellensatz…
The automated generation of exercises may substantially reduce the time educators devote to manual exercise design. A major obstacle to the integration of such automation into teaching practice, however, lies in the ability to control the…
This paper is concerned with certifying that a given point is near an exact root of an overdetermined or singular polynomial system with rational coefficients. The difficulty lies in the fact that consistency of overdetermined systems is…
This paper investigates the minimization of the expectation of piecewise polynomial loss functions over Wasserstein balls. This optimization problem often appears as a key sub-problem of distributionally robust optimization problems. We…
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…
An emerging branch of control theory specialises in certificate learning, concerning the specification of a desired (possibly complex) system behaviour for an autonomous or control model, which is then analytically verified by means of a…
Using polynomial equations to model combinatorial problems has been a popular tool both in computational combinatorics as well as an approach to proving new theorems. In this paper, we look at several combinatorics problems modeled by…
Quantum optimal control plays a crucial role in the development of quantum technologies, particularly in the design and implementation of fast and accurate gates for quantum computing. Here, we present a method to synthesize gates using the…
Symmetry and dominance breaking can be crucial for solving hard combinatorial search and optimisation problems, but the correctness of these techniques sometimes relies on subtle arguments. For this reason, it is desirable to produce…
We study separable plus quadratic (SPQ) polynomials, i.e., polynomials that are the sum of univariate polynomials in different variables and a quadratic polynomial. Motivated by the fact that nonnegative separable and nonnegative quadratic…
Recently, non-SOS Positivstellens\"atze for polynomials on compact semialgebraic sets, following the general form of Schm\"{u}dgen's Positivstellensatz, have been derived by appropriately replacing the SOS polynomials with other classes of…
We consider the problem of minimizing a polynomial $f$ over the binary hypercube. We show that, for a specific set of polynomials, their binary non-negativity can be checked in a polynomial time via minimum cut algorithms, and we construct…
We study $\textit{sparse singular value certificates}$ for random rectangular matrices. If $M$ is an $n \times d$ matrix with independent Gaussian entries, we give a new family of polynomial-time algorithms which can certify upper bounds on…
Polynomial meshes (called sometimes "norming sets") allow us to estimate the supremum norm of polynomials on a fixed compact set by the norm on its discrete subset. We give a general construction of polynomial weakly admissible meshes on…