Section: New Results

Solving Systems over the Reals and Applications

Answering connectivity queries in real algebraic sets

A roadmap for a semi-algebraic set $S$ is a curve which has a non-empty and connected intersection with all connected components of $S$. Hence, this kind of object, introduced by Canny, can be used to answer connectivity queries (with applications, for instance, to motion planning) but has also become of central importance in effective real algebraic geometry, since it is used in higher-level algorithms. In [10], we provide a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets. Its output size and running time are polynomial in ${\left(nD\right)}^{nlogd}$, where $D$ is the maximum of the degrees of the input polynomials, $d$ is the dimension of the set under consideration and n is the number of variables. More precisely, the running time of the algorithm is essentially subquadratic in the output size. Even under our assumptions, it is the first roadmap algorithm with output size and running time polynomial in ${\left(nD\right)}^{nlogd}$.

Polynomial optimization and semi-definite programming

In [6], we describe our freely distributed Maple library spectra, for Semidefinite Programming solved Exactly with Computational Tools of Real Algebra. It solves linear matrix inequalities, a fundamental object in effective real algebraic geometry and polynomial optimization, with symbolic computation in exact arithmetic and it is targeted to small-size, possibly degenerate problems for which symbolic infeasibility or feasibility certificates are required.

The positive semidefinite rank of a convex body $C$ is the size of its smallest positive semi-definite formulation. In [5], we show that the positive semidefinite rank of any convex body $C$ is at least $\sqrt{log\left(d\right)}$ where $d$ is the smallest degree of a polynomial that vanishes on the boundary of the polar of $C$. This improves on the existing bound which relies on results from quantifier elimination. Our proof relies on the Bézout bound applied to the Karush-Kuhn-Tucker conditions of optimality. We discuss the connection with the algebraic degree of semidefinite programming and show that the bound is tight (up to constant factor) for random spectrahedra of suitable dimension.

The Complexity of an Adaptive Subdivision Method for Approximating Real Curves

In [14] we present the first complexity analysis of the algorithm by Plantinga and Vegter for approximating real implicit curves and surfaces. This approximation algorithm certifies the topological correctness of the output using both subdivision and interval arithmetic. In practice, it has been seen to be quite efficient; our goal is to quantify this efficiency. We focus on the subdivision step (and not the approximation step) of the Plantinga and Vegter algorithm. We begin by extending the subdivision step to arbitrary dimensions. We provide a priori worst-case bounds on the complexity of this algorithm both in terms of the number of subregions constructed and the bit complexity for the construction. Then, we use continuous amortization to derive adaptive bounds on the complexity of the subdivided region. We also provide examples showing our bounds are tight.