Related papers: Imaginary projections of polynomials

This article studies optional and predictable projections of integrands and convex-valued stochastic processes. The existence and uniqueness are shown under general conditions that are analogous to those for conditional expectations of…

We study a properly convex real projective manifold with (possibly empty) compact, strictly convex boundary, and which consists of a compact part plus finitely many convex ends. We extend a theorem of Koszul which asserts that for a compact…

We prove a new Elekes-Szab\'o type estimate on the size of the intersection of a Cartesian product $A\times B\times C$ with an algebraic surface $\{f=0\}$ over the reals. In particular, if $A,B,C$ are sets of $N$ real numbers and $f$ is a…

We study algebraic dynamical systems (and, more generally, $\sigma$-varieties) $\Phi:{\mathbb A}^n_{\mathbb C} \to {\mathbb A}^n_{\mathbb C}$ given by coordinatewise univariate polynomials by refining a theorem of Ritt. More precisely, we…

In this work we prove constructively that the complement ${\mathbb R}^n\setminus{\mathcal K}$ of an $n$-dimensional unbounded convex polyhedron ${\mathcal K}\subset{\mathbb R}^n$ and the complement ${\mathbb R}^n\setminus{\rm Int}({\mathcal…

In this note, we provide explicit expressions for the projections onto the graph of a quadratic polynomial. The projections are obtained by examining the critical points of the associated quartic polynomial, that is, the roots of the cubic…

We study the projection onto the set of feasible inputs and the set of feasible solutions of a polynomial optimisation problem (POP). Our motivation is increasing the robustness of solvers for POP: Without a priori guarantees of feasibility…

We investigate structural properties of the cone of roots of relative Steiner polynomials of convex bodies. We prove that they are closed, monotonous with respect to the dimension, and that they cover the whole upper half-plane, except the…

Consider a finite collection of affine hyperplanes in $\mathbb R^d$. The hyperplanes dissect $\mathbb R^d$ into finitely many polyhedral chambers. For a point $x\in \mathbb R^d$ and a chamber $P$ the metric projection of $x$ onto $P$ is the…

We investigate projection constants within classes of multivariate polynomials over finite-dimensional real Hilbert spaces. Specifically, we consider the projection constant for spaces of spherical harmonics and spaces of homogeneous…

In this paper, we study polynomial norms, i.e. norms that are the $d^{\text{th}}$ root of a degree-$d$ homogeneous polynomial $f$. We first show that a necessary and sufficient condition for $f^{1/d}$ to be a norm is for $f$ to be strictly…

We show that the multipole vector decomposition, recently introduced by Copi et al., is a consequence of Sylvester's theorem, and corresponds to the Maxwell's representation. Analyzing it in terms of harmonic polynomials, we show that this…

We prove a sharp upper bound for the projective dimension of ideals of height two generated by quadrics in a polynomial ring with arbitrary large number of variables.

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size, and its roots are called {\em independence roots}. We investigate the stability of such polynomials, that is, conditions…

We strengthen the Weierstrass approximation theorem by proving that any real-valued continuous function on an interval $I \subset \mathbb{R}$ can be uniformly approximated by a real-valued polynomial whose only (possibly complex) critical…

In this paper we study the affine geometric structure of the graph of a polynomial $f \in \mathbb{R} [x,y]$. We provide certain criteria to determine when the parabolic curve is compact and when the unbounded component of its complement is…

We consider the numerical evaluation of one dimensional projections of general multivariate stable densities introduced by Abdul-Hamid and Nolan (1998). In their approach higher order derivatives of one dimensional densities are used, which…

We study the Hopf monoid of convex geometries, which contains partial orders as a Hopf submonoid, and investigate the combinatorial invariants arising from canonical characters. Each invariant consists of a pair: a polynomial and a more…

We show that (as conjectured by Lin and Wang) when a Vassiliev invariant of type $m$ is evaluated on a knot projection having $n$ crossings, the result is bounded by a constant times $n^m$. Thus the well known analogy between Vassiliev…

The long-standing problem of minimal projections is addressed from a computational point of view. Techniques to determine bounds on the projection constants of univariate polynomial spaces are presented. The upper bound, produced by a…