Related papers: Complexity and speed of semi-algebraic multi-persi…

We give an algorithm with singly exponential complexity for computing the barcodes up to dimension $\ell$ (for any fixed $\ell \geq 0$) of the filtration of a given semi-algebraic set by the sub-level sets of a given polynomial. Our…

We prove that for any $\ell \geq 0$, there exists an algorithm which takes as input a description of a semi-algebraic subset $S \subset \mathbb{R}^k$ given by a quantifier-free first order formula $\phi$ in the language of the reals, and…

We give new positive and negative results (some conditional) on speeding up computational algebraic geometry over the reals: (1) A new and sharper upper bound on the number of connected components of a semialgebraic set. Our bound is novel…

For any $\ell > 0$, we present an algorithm which takes as input a semi-algebraic set, $S$, defined by $P_1 \leq 0,...,P_s \leq 0$, where each $P_i \in \R[X_1,...,X_k]$ has degree $\leq 2,$ and computes the top $\ell$ Betti numbers of $S$,…

Let $\mathrm{R}$ be a real closed field and $\mathrm{C}$ the algebraic closure of $\mathrm{R}$. We give an algorithm for computing a semi-algebraic basis for the first homology group, $\mathrm{H}_1(S,\mathbb{F})$, with coefficients in a…

Let $\mathfrak p$ be a proper parabolic subalgebra of a simple Lie algebra $\mathfrak g$. Writing $\mathfrak p=\mathfrak r\oplus \mathfrak m$, with $\mathfrak r$ being the Levi factor of $\mathfrak p$ and $\mathfrak m$ the nilpotent radical…

Let $\R$ be a real closed field, $ {\mathcal Q} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k], $ with $ \deg_{Y}(Q) \leq 2, \deg_{X}(Q) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m$, and $ {\mathcal P} \subset \R[X_1,...,X_k] $ with $\deg_{X}(P)…

We prove that the number of distinct homotopy types of limits of one-parameter semi-algebraic families of closed and bounded semi-algebraic sets is bounded singly exponentially in the additive complexity of any quantifier-free first order…

We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in $\R^\ell$, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but…

In this paper we describe an algorithm that takes as input a description of a semi-algebraic set $S \subset \R^k$, defined by a Boolean formula with atoms of the form $P > 0, P < 0, P=0$ for $P \in {\mathcal P} \subset \R[X_1,...,X_k],$ and…

We provide a numerical scheme to approximate as closely as desired the Gaussian or exponential measure $\mu(\om)$ of (not necessarily compact) basic semi-algebraic sets$\om\subset\R^n$. We obtain two monotone (non increasing and non…

Real algebraic geometry is the study of semi-algebraic sets, subsets of $\R^k$ defined by Boolean combinations of polynomial equalities and inequalities. The focus of this thesis is to study quantitative results in real algebraic geometry,…

We present an algorithm which takes as input a closed semi-algebraic set, $S \subset \R^k$, defined by \[ P_1 \leq 0, ..., P_\ell \leq 0, P_i \in \R[X_1,...,X_k], \deg(P_i) \leq 2, \] and computes the Euler-Poincar\'e characteristic of $S$.…

Given a polynomial $f$ and a semi-algebraic set $S$, we provide a symbolic algorithm to find the equations and inequalities defining a semi-algebraic set $Q$ which is identical to the closure of the image of $S$ under $f$, i.e.,…

We consider the asymmetric simple exclusion process (ASEP) on a semi-infinite chain which is coupled at the end to a reservoir with a particle density that changes periodically in time. It is shown that the density profile assumes a…

Developing an algorithm for computing the Betti numbers of semi-algebraic sets with singly exponential complexity has been a holy grail in algorithmic semi-algebraic geometry and only partial results are known. In this paper we consider the…

We show that the closed convex hull of any one-dimensional semi-algebraic subset of R^n has a semidefinite representation, meaning that it can be written as a linear projection of the solution set of some linear matrix inequality. This is…

Probabilistic inference in pairwise Markov Random Fields (MRFs), i.e. computing the partition function or computing a MAP estimate of the variables, is a foundational problem in probabilistic graphical models. Semidefinite programming…

We show that a certain tensor norm, the completely bounded norm, can be expressed by a semidefinite program. This tensor norm recently attracted attention in the field of quantum computing, where it was used by Arunachalam, Bri\"{e}t and…

We generalize the geometric sequence $\{a^p, a^{p-1}b, a^{p-2}b^2,...,b^p\}$ to allow the $p$ copies of $a$ (resp. $b$) to all be different. We call the sequence $\{a_1a_2a_3\cdots a_p, b_1a_2a_3\cdots a_p, b_1b_2a_3\cdots a_p,\ldots,…