Related papers: Computing the Top Betti Numbers of Semi-algebraic …

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)…

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…

Let $S \subset \R^{k + m}$ be a compact semi-algebraic set defined by a system of $\ell$ polynomial inequalities of degree at most 2. $ Let $\pi$ denote the standard projection from $\R^{k + m}$ onto $\R^m$. We prove that for any $q >0$,…

In this paper we consider the problem of bounding the Betti numbers, $b_i(S)$, of a semi-algebraic set $S \subset \R^k$ defined by polynomial inequalities $P_1 \geq 0,...,P_s \geq 0$, where $P_i \in \R[X_1,...,X_k]$ and $\deg(P_i) \leq 2$,…

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 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…

Let $\mathrm{R}$ be a real closed field. The problem of obtaining tight bounds on the Betti numbers of semi-algebraic subsets of $\mathrm{R}^k$ in terms of the number and degrees of the defining polynomials has been an important problem in…

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 $S$.…

In this thesis, we consider semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials. Semi-algebraic sets of $R^k$ are defined as the smallest family of sets in $R^k$ that contains the algebraic sets as well as the…

In this paper we describe a singly exponential algorithm for computing the first Betti number of a given semi-algebraic set. Singly exponential algorithms for computing the zero-th Betti number, and the Euler-Poincar\'e characteristic, were…

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…

Let $\mathrm{R}$ be a real closed field. We prove that for any fixed $d$, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $\mathrm{R}^k$ defined by polynomials of degrees bounded by $d$ vanishes in…

A semi-algebraic set is a subset of $\mathbb{R}^n$ defined by a finite collection of polynomial equations and inequalities. In this paper, we investigate the problem of determining whether two points in such a set belong to the same…

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 describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of semialgebraic sets given by Boolean formulas. The algorithm works in weak exponential time. This means that outside a…

It is known that point searching in basic semialgebraic sets and the search for globally minimal points in polynomial optimization tasks can be carried out using $(s\,d)^{O(n)}$ arithmetic operations, where $n$ and $s$ are the numbers of…

We looked into the algorithm for calculating Betti numbers presented by Lloyd, Garnerone, and Zanardi (LGZ). We present a new algorithm in the same spirit as LGZ with the intent of clarifying quantum algorithms for computing Betti numbers.…

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of closed semialgebraic sets given by Boolean formulas without negations over lax polynomial inequalities. The algorithm works in weak…

In this paper, an exact algorithm in polynomial time is developed to solve unrestricted binary quadratic programs. The computational complexity is $O\left( n^{\frac{15}{2}}\right) $, although very conservative, it is sufficient to prove…

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,…