Related papers: Constant-depth circuits for polynomial GCD over an…

In this paper we give an Immerman's Theorem for real-valued computation. We define circuits operating over real numbers and show that families of such circuits of polynomial size and constant depth decide exactly those sets of vectors of…

We present a new algorithm for isolating the real roots of a system of multivariate polynomials, given in the monomial basis. It is inspired by existing subdivision methods in the Bernstein basis; it can be seen as generalization of the…

We investigate the symmetry component of the center variety of polynomial differential systems, corresponding to systems with an axis of symmetry in the real plane. We give a general algorithm to find this irreducible subvariety and compute…

We establish a connection between continuous-variable quantum computing and high-dimensional integration by showing that the outcome probabilities of continuous-variable instantaneous quantum polynomial (CV-IQP) circuits are given by…

Let $g(X)$ be a polynomial over a finite field ${\mathbb F}_q$ with degree $o(q^{1/2})$, and let $\chi$ be the quadratic residue character. We give a polynomial time algorithm to recover $g(X)$ (up to perfect square factors) given the…

The problem of characterizing graphs by their generalized spectra has received significant attention in recent years. This paper provides a complete proof of a conjecture proposed by Wang, Wang, and Zhu (European J. Combin., 2023), which…

The paper describes the algebraic structure of the graded algebra of differentially homogeneous polynomials of fixed finite order. We show that it is a finitely generated algebra, and we exhibit a minimal set of generators. Along the way,…

Differential (Ore) type polynomials with "approximate" polynomial coefficients are introduced. These provide an effective notion of approximate differential operators, with a strong algebraic structure. We introduce the approximate Greatest…

For a native gate set which includes all single-qubit gates, we apply results from symplectic geometry to analyze the spaces of two-qubit programs accessible within a fixed number of gates. These techniques yield an explicit description of…

This article characterizes the rank-one factorization of auto-correlation matrix polynomials. We establish a sufficient and necessary uniqueness condition for uniqueness of the factorization based on the greatest common divisor (GCD) of…

We generalize the classical theory of periodic continued fractions (PCFs) over ${\mathbf Z}$ to rings ${\mathcal O}$ of $S$-integers in a number field. Let ${\mathcal B}=\{\beta, {\beta^*}\}$ be the multi-set of roots of a quadratic…

Motivated by a rigidity-theoretic perspective on the Localization Problem in 2D, we develop an algorithm for computing circuit polynomials in the algebraic rigidity matroid associated to the Cayley-Menger ideal for $n$ points in 2D. We…

In this paper, we consider a family of closed planar algebraic curves $\mathcal{C}$ which are given in parametrization form via a trigonometric polynomial $p$. When $\mathcal{C}$ is the boundary of a compact convex set, the polynomial $p$…

Over fields of characteristic unequal to $2$, we can identify symmetric matrices with homogeneous polynomials of degree $2$. This allows us to view symmetric rank-metric codes as living inside the space of such polynomials. In this paper,…

We show that tools from circuit complexity can be used to study decompositions of global constraints. In particular, we study decompositions of global constraints into conjunctive normal form with the property that unit propagation on the…

Let $G$ be a Cameron--Walker graph on $n$ vertices and $J_G$ the binomial edge ideal of $G$. Let $S$ denote the polynomial ring in $2n$ variables over a field. It is shown that the following conditions are equivalent: (i) $S/J_G$ is…

The Generalized Persistence Diagram (GPD) for multi-parameter persistence naturally extends the classical notion of persistence diagram for one-parameter persistence. However, unlike its classical counterpart, computing the GPD remains a…

There are many different algebraic, geometric and combinatorial objects that one can attach to a complex polynomial with distinct roots. In this article we introduce a new object that encodes many of the existing objects that have…

We propose a mathematical walk around the gcd of the values $A(n)$ and $B(n)$ of two polynomials evaluated at an integer $n$. This is an opportunity to use a very powerful tool: the resultant.

In [19], Taslaman, Tisseur, and Zaballa show that any regular matrix polynomial $P(\lambda)$ over an algebraically closed field is spectrally equivalent to a triangular matrix polynomial of the same degree. When $P(\lambda)$ is real and…