arXiv subjects

Michael Kettner

Publications and source records attributed to Michael Kettner.

3 recordsLinked to original sources

arXiv2007

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

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.

math.AG↗

arXiv2007

Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

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 polynomial in $\ell$. More precisely, we prove the following. Let $\R$ be a real closed field and let \[ {\mathcal P} = \{P_1,...,P_m\} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k], \] with ${\rm deg}_Y(P_i) \leq 2, {\rm deg}_X(P_i) \leq d, 1 \leq i \leq m$. Let $S \subset \R^{\ell+k}$ be a semi-algebraic set, defined by a Boolean formula without negations, whose atoms are of the form, $P \geq 0, P\leq 0, P \in {\mathcal P}$. Let $π: \R^{\ell+k} \to \R^k$ be the projection on the last k co-ordinates. Then, the number of stable homotopy types amongst the fibers $S_{\x} = π^{-1}(\x) \cap S$ is bounded by \[ (2^m\ell k d)^{O(mk)}. \]

math.AG↗

arXiv2006

A sharper estimate on the Betti numbers of sets defined by quadratic inequalities

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 $°(P_i) \leq 2$, for $1 \leq i \leq s$. We prove that for $0\le i\le k-1$, \[ b_i(S) \le{1/2}(\sum_{j=0}^{min\{s,k-i\}}{{s}\choose j}{{k+1}\choose {j}}2^{j}). \] In particular, for $2\le s\le \frac{k}{2}$, we have \[ b_i(S)\le {1/2} 3^{s}{{k+1}\choose {s}} \leq {1/2} (\frac{3e(k+1)}{s})^s. \] This improves the bound of $k^{O(s)}$ proved by Barvinok. This improvement is made possible by a new approach, whereby we first bound the Betti numbers of non-singular complete intersections of complex projective varieties defined by generic quadratic forms, and use this bound to obtain bounds in the real semi-algebraic case.

math.AG↗