Algorithmic and topological aspects of semi-algebraic sets defined by quadratic polynomial

TL;DR

This thesis advances the understanding of semi-algebraic sets defined by quadratic polynomials by providing new bounds on topological invariants and developing efficient algorithms for computing their properties.

Contribution

It introduces improved bounds on Betti numbers and homotopy types, and presents two novel algorithms for analyzing semi-algebraic sets defined by quadratic polynomials.

Findings

New bounds on Betti numbers and stable homotopy types

An efficient algorithm for computing connected components and Betti numbers

A semi-numerical method for intersecting quadratic surfaces

Abstract

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 sets defined by polynomial inequalities, and which is also closed under the boolean operations (complementation, finite unions and finite intersections). We prove new bounds on the Betti numbers as well as on the number of different stable homotopy types of certain fibers of semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials, in terms of the parameters of the system of polynomials defining them, which improve the known results. We conclude the thesis with presenting two new algorithms along with their implementations. The first algorithm computes the number of connected components and the first Betti number of a semi-algebraic set defined by compact objects in $\mathbb{R}^k$ which are simply connected. This algorithm improves the well-know method using a triangulation of the semi-algebraic set. Moreover, the algorithm has been efficiently implemented which was not possible before. The second algorithm computes efficiently the real intersection of three quadratic surfaces in $\mathbb{R}^3$ using a semi-numerical approach.