Section: New Results

Solving Systems over the Reals and Applications

On the Boolean complexity of real root refinement

In [32] we assume that a real square-free polynomial $A$ has a degree $d$, a maximum coefficient bitsize $\tau$ and a real root lying in an isolating interval and having no nonreal roots nearby (we quantify this assumption). Then, we combine the Double Exponential Sieve algorithm (also called the Bisection of the Exponents), the bisection, and Newton iteration to decrease the width of this inclusion interval by a factor of $t=2^{-L}$. The algorithm has Boolean complexity ${\tilde{O}}_B(d^2\tau +dL)$. Our algorithms support the same complexity bound for the refinement of $r$ roots, for any $r\le d$.

On the minimum of a polynomial function on a basic closed semialgebraic set and applications

In [17] we give an explicit upper bound for the algebraic degree and an explicit lower bound for the absolute value of the minimum of a polynomial function on a compact connected component of a basic closed semialgebraic set when this minimum is not zero. We also present extensions of these results to non-compact situations. As an application, we obtain a lower bound for the separation of two disjoint connected components of basic closed semialgebraic sets, when at least one of them is compact.

Rational solutions to Linear Matrix Inequalities and Sums of Squares

Consider a $(D\times D)$ symmetric matrix $𝖠$ whose entries are linear forms in $\mathbb{Q}[X_1,...,X_k]$ with coefficients of bit size $\le \tau$. In [31] , we provide an algorithm which decides the existence of rational solutions to the linear matrix inequality $𝖠⪰0$ and outputs such a rational solution if it exists. This problem is of first importance: it can be used to compute algebraic certificates of positivity for multivariate polynomials. Our algorithm runs within ${\left(k\tau \right)}^{O\left(1\right)}{2}^{O(min(k,D)D^2)}{D}^{O\left(D^2\right)}$ bit operations; the bit size of the output solution is dominated by ${\tau }^{O\left(1\right)}{2}^{O(min(k,D)D^2)}$. These results are obtained by designing algorithmic variants of constructions introduced by Klep and Schweighofer. This leads to the best complexity bounds for deciding the existence of sums of squares with rational coefficients of a given polynomial. We have implemented the algorithm; it has been able to tackle Scheiderer's example of a multivariate polynomial that is a sum of squares over the reals but not over the rationals; providing the first computer validation of this counter-example to Sturmfels' conjecture.

Exact Voronoi diagram of smooth convex pseudo-circles: General predicates, and implementation for ellipses

In [10] we examine the problem of computing exactly the Voronoi diagram (via the dual Delaunay graph) of a set of, possibly intersecting, smooth convex pseudo-cirlces in the Euclidean plane, given in parametric form. Pseudo-circles are (convex) sites, every pair of which has at most two intersecting points. The Voronoi diagram is constructed incrementally. Our first contribution is to propose robust and efficient algorithms, under the exact computation paradigm, for all required predicates, thus generalizing earlier algorithms for non-intersecting ellipses. Second, we focus on InCircle , which is the hardest predicate, and express it by a simple sparse $5\times 5$ polynomial system, which allows for an efficient implementation by means of successive Sylvester resultants and a new factorization lemma. The third contribution is our CGAL-based C++ software for the case of possibly intersecting ellipses, which is the first exact implementation for the problem. Our code spends about a minute to construct the Voronoi diagram of 200 ellipses, when few degeneracies occur. It is faster than the CGAL segment Voronoi diagram, when ellipses are approximated by $k$-gons for $k>15$, and a state-of-the-art implementation of the Voronoi diagram of points, when each ellipse is approximated by more than 1250 points.

Patience of Matrix Games

In [15] , for matrix games we study how small nonzero probability must be used in optimal strategies. We show that for $n\times n$ win-lose-draw games (i.e. $(-1,0,1)$ matrix games) nonzero probabilities smaller than $n^{-O\left(n\right)}$ are never needed. We also construct an explicit $n\times n$ win-lose game such that the unique optimal strategy uses a nonzero probability as small as $n^{-\Omega \left(n\right)}$. This is done by constructing an explicit $(-1,1)$ nonsingular $n\times n$ matrix, for which the inverse has only nonnegative entries and where some of the entries are of value $n^{\Omega \left(n\right)}$.

A polynomial approach for extracting the extrema of a spherical function and its application in diffusion MRI

Antipodally symmetric spherical functions play a pivotal role in diffusion MRI in representing sub-voxel-resolution microstructural information of the underlying tissue. This information is described by the geometry of the spherical function. In [14] we propose a method to automatically compute all the extrema of a spherical function. We then classify the extrema as maxima, minima and saddle-points to identify the maxima. We take advantage of the fact that a spherical function can be described equivalently in the spherical harmonic (SH) basis, in the symmetric tensor (ST) basis constrained to the sphere, and in the homogeneous polynomial (HP) basis constrained to the sphere. We extract the extrema of the spherical function by computing the stationary points of its constrained HP representation. Instead of using traditional optimization approaches, which are inherently local and require exhaustive search or re-initializations to locate multiple extrema, we use a novel polynomial system solver which analytically brackets all the extrema and refines them numerically, thus missing none and achieving high precision. To illustrate our approach we consider the Orientation Distribution Function (ODF). In diffusion MRI the ODF is a spherical function which represents a state-of-the-art reconstruction algorithm whose maxima are aligned with the dominant fiber bundles. It is, therefore, vital to correctly compute these maxima to detect the fiber bundle directions. To demonstrate the potential of the proposed polynomial approach we compute the extrema of the ODF to extract all its maxima. This polynomial approach is, however, not dependent on the ODF and the framework presented in this line of work can be applied to any spherical function described in either the SH basis, ST basis or the HP basis.

Improving Angular Speed Uniformity by Reparameterization

In [20] we introduce the notion of angular speed uniformity as a quality measure for parameter-izations of plane curves and propose an algorithm to compute uniform reparameterizations for quadratic and cubic curves. We prove that only straight lines have uniform rational parameterizations. For any plane curve other than lines, we show how to find a rational reparameterization that has the maximum uniformity among all the rational parameterizations of the same degree. We also establish specific results for quadratic and certain cubic Bézier curves.

Formalization and Specification of Geometric Knowledge Objects

[7] presents our work on the identification, formalization, structuring, and specification of geometric knowledge objects for the purpose of semantic representation and knowledge management. We classify geometric knowledge according to how it has been accumulated and represented in the geometric literature, formalize geometric knowledge statements by adapting the language of first-order logic, specify knowledge objects with embedded knowledge in a retrievable and extensible data structure, and organize them by modeling the hierarchic structure of relations among them. Some examples of formal specification for geometric knowledge objects are given to illustrate our approach. The underlying idea of the approach has been used successfully for automated geometric reasoning, knowledge base creation, and electronic document generation.

A Framework for Improving Uniformity of Parameterizations of Curves

In [16] we define quasi-speed as a generalization of linear speed and angular speed for parameterizations of curves and use the uniformity of quasi-speed to measure the quality of the parameterizations. With such conceptual setting, a general framework is developed for studying uniformity behaviors under reparameterization via proper parameter transformation and for computing reparameterizations with improved uniformity of quasispeed by means of optimal single-piece, $C^0$ piecewise, and $C^1$ piecewise Möbius transformations. Algorithms are described for uniformity-improved reparameterization using different Möbius transformations with different optimization techniques. Examples are presented to illustrate the concepts, the framework, and the algorithms. Experimental results are provided to validate the framework and to show the efficiency of the algorithms.