## Section: New Results

### Non-Linear Computational Geometry

Participants : Laurent Dupont, Nuwan Herath Mudiyanselage, George Krait, Sylvain Lazard, Viviane Ledoux, Guillaume Moroz, Marc Pouget.

#### Clustering Complex Zeros of Triangular Systems of Polynomials

This work, presented at the CASC'19 Conference [23], gives the first algorithm for finding a set of natural $\u03f5$-clusters of complex zeros of a regular triangular system of polynomials within a given polybox in ${\u2102}^{n}$, for any given $\u03f5>0$. Our algorithm is based on a recent near-optimal algorithm of Becker et al (2016) for clustering the complex roots of a univariate polynomial where the coefficients are represented by number oracles. Our algorithm is based on recursive subdivision. It is local, numeric, certified and handles solutions with multiplicity. Our implementation is compared to well-known homotopy solvers on various triangular systems. Our solver always gives correct answers, is often faster than the homotopy solvers that often give correct answers, and sometimes faster than the ones that give sometimes correct results.

*In collaboration with R. Imbach and C. Yap (Courant Institute of
Mathematical Sciences, New York University, USA).*

#### Numerical Algorithm for the Topology of Singular Plane Curves

We are interested in computing the topology of plane singular curves. For this, the singular points must be isolated. Numerical methods for isolating singular points are efficient but not certified in general. We are interested in developing certified numerical algorithms for isolating the singularities. In order to do so, we restrict our attention to the special case of plane curves that are projections of smooth curves in higher dimensions. In this setting, we show that the singularities can be encoded by a regular square system whose isolation can be certified by numerical methods. This type of curves appears naturally in robotics applications and scientific visualization. This work was presented at the EuroCG'19 Conference [24].

#### Reliable Computation of the Singularities of the Projection in ${\mathbb{R}}^{3}$ of a Generic Surface of ${\mathbb{R}}^{4}$

Computing efficiently the singularities of surfaces embedded in ${\mathbb{R}}^{3}$ is a difficult problem, and most state-of-the-art approaches only handle the case of surfaces defined by polynomial equations. Let $F$ and $G$ be ${C}^{\infty}$ functions from ${\mathbb{R}}^{4}$ to $\mathbb{R}$ and $\mathcal{M}=\{(x,y,z,t)\in {\mathbb{R}}^{4}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}F(x,y,z,t)=G(x,y,z,t)=0\}$ be the surface they define. Generically, the surface $\mathcal{M}$ is smooth and its projection $\Omega $ in ${\mathbb{R}}^{3}$ is singular. After describing the types of singularities that appear generically in $\Omega $, we design a numerically well-posed system that encodes them. This can be used to return a set of boxes that enclose the singularities of $\Omega $ as tightly as required. As opposed to state-of-the art approaches, our approach is not restricted to polynomial mappings, and can handle trigonometric or exponential functions for example. This work was presented at the MACIS'19 Conference [19].

*In collaboration with Sény Diatta (University Assane Seck of Ziguinchor,
Senegal)*

#### Evaluation of Chebyshev polynomials on intervals and application to root finding

In approximation theory, it is standard to approximate functions by polynomials expressed in the Chebyshev basis. Evaluating a polynomial $f$ of degree $n$ given in the Chebyshev basis can be done in $O\left(n\right)$ arithmetic operations using the Clenshaw algorithm. Unfortunately, the evaluation of $f$ on an interval $I$ using the Clenshaw algorithm with interval arithmetic returns an interval of width exponential in $n$. We describe a variant of the Clenshaw algorithm based on ball arithmetic that returns an interval of width quadratic in $n$ for an interval of small enough width. As an application, our variant of the Clenshaw algorithm can be used to design an efficient root finding algorithm. This work was presented at the MACIS'19 Conference [21].

#### Using Maple to analyse parallel robots

We present the SIROPA Maple Library which has been designed to study serial and parallel manipulators at the conception level. We show how modern algorithms in Computer Algebra can be used to study the workspace, the joint space but also the existence of some physical capabilities w.r.t. to some design parameters left as degree of freedom for the designer of the robot. This work was presented at the Maple Conference 2019 [18].

*In collaboration with Philippe Wenger, Damien Chablat
(Laboratoire des Sciences du Numérique de Nantes, UMR CNRS 6004)
and Fabrice Rouillier (project team *
Ouragan
*)*