Related papers: Lower bounds for polynomials using geometric progr…

We present a branch-and-bound algorithm to improve the lower bounds obtained by SONC/SAGE. The running time is fixed-parameter tractable in the number of variables. Furthermore, we describe a new heuristic to obtain a candidate for the…

We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new…

This paper derives new algorithms for signomial programming, a generalization of geometric programming. The algorithms are based on a generic principle for optimization called the MM algorithm. In this setting, one can apply the…

The breakthrough paper of Croot, Lev, Pach \cite{CLP} on progression-free sets in $\Z_4^n$ introduced a polynomial method that has generated a wealth of applications, such as Ellenberg and Gijswijt's solutions to the cap set problem…

Putinar's Positivstellensatz is a central theorem in real algebraic geometry. It states the following: If you have a set $S= \{ x \in R^n \ | \ g_1 (x) \geq 0, ... , g_m(x) \geq 0\}$ described by some real polynomials $g_i$, then every real…

Generalised planning (GP) refers to the task of synthesising programs that solve families of related planning problems. We introduce a novel, yet simple method for GP: given a set of training problems, for each problem, compute an optimal…

We consider the semi-infinite optimization problem: $f^*:=\min_{x\in X}\:\{f(x): g(x,y)\,\leq \,0,\:\forally\in Y_x\}$, where $f,g$ are polynomials and $X\subset R^n$ as well as $Y_\x\subset R^p$, $x\in X$, are compact basic semi-algebraic…

Our contribution in this paper is two folded. We consider first the case of linear programming with real coefficients and give a method which allows the computation of a new upper bound on the distance from the origin to a feasible point.…

The approximate degree of a Boolean function $f(x_{1},x_{2},\ldots,x_{n})$ is the minimum degree of a real polynomial that approximates $f$ pointwise within $1/3$. Upper bounds on approximate degree have a variety of applications in…

In recent years, copositive programming has received significant attention for its ability to model hard problems in both discrete and continuous optimization. Several relaxations of copositive programs based on semidefinite programming…

Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, %% on these polynomials, we prove that the optimality conditions always hold on…

A degree-$d$ polynomial $p$ in $n$ variables over a field $\F$ is {\em equidistributed} if it takes on each of its $|\F|$ values close to equally often, and {\em biased} otherwise. We say that $p$ has a {\em low rank} if it can be expressed…

In analogy with the regularity lemma of Szemer\'edi, regularity lemmas for polynomials shown by Green and Tao (Contrib. Discrete Math. 2009) and by Kaufman and Lovett (FOCS 2008) modify a given collection of polynomials \calF =…

Let $f_1(x),\ldots,f_n(x)$ be some polynomials. The upper bound on the number of $x\in\mathbb F_p$ such that $f_1(x),\ldots,f_n(x)$ are roots of unit of order $t$ is obtained. This bound generalize the bound of the paper \cite{V-S} to the…

In this paper, we present a new method for computing bounded-degree factors of lacunary multivariate polynomials. In particular for polynomials over number fields, we give a new algorithm that takes as input a multivariate polynomial f in…

For each $n$, let $\text{RD}(n)$ denote the minimum $d$ for which there exists a formula for the general polynomial of degree $n$ in algebraic functions of at most $d$ variables. In 1945, Segre called for a better understanding of the large…

Let $\xx_n=(x_1,\ldots,x_n)$ and $f\in \R[\xx_n,k]$. The problem of finding all $k_0$ such that $f(\xx_n,k_0)\ge 0$ on $\mathbb{R}^n$ is considered in this paper, which obviously takes as a special case the problem of computing the global…

Motivated by applications in robotics and computer vision, we study problems related to spatial reasoning of a 3D environment using sublevel sets of polynomials. These include: tightly containing a cloud of points (e.g., representing an…

The motivation of this work stems from the numerical approximation of bounded functions by polynomials satisfying the same bounds. The present contribution makes use of the recent algebraic characterization found in [B. Despr\'es, Numer.…

It is well-known that any sum of squares (SOS) program can be cast as a semidefinite program (SDP) of a particular structure and that therein lies the computational bottleneck for SOS programs, as the SDPs generated by this procedure are…