arXiv subjects

Marc Pouget

Publications and source records attributed to Marc Pouget.

6 recordsLinked to original sources

arXiv2026

The Hyperbolic Surface Distance, Diameter, and Dirichlet Problems

Despite the prominence of hyperbolic surfaces in mathematics, basic algorithmic questions about them, even computing the distance between two points, have remained open, leaving many features of these surfaces inaccessible. The classical machinery assumes a polyhedral structure absent on a smooth surface. We remove these obstacles. We begin with an efficient $O(g^2)$ algorithm for the distance between two points, where $g$ is the genus of the surface. Building on it, we obtain an $O(g^2 \log g)$ method for answering distance queries from a fixed source and, as a consequence, for recentering a Dirichlet domain around an arbitrary point. This understanding of distances on the surface then lets us approximate the diameter to within any $\eps$ in time $O(g^3 \log g / \eps^2)$. We further show that the diameter, a single real number encoding a great deal about the surface, is exactly computable. Its hyperbolic cosine is an algebraic number over the field encoding the coefficients of the hyperbolic isometries defining the surface.

cs.CG↗

arXiv2018

Clustering Complex Zeros of Triangular Systems of Polynomials

This paper gives the first algorithm for finding a set of natural $\epsilon$-clusters of complex zeros of a triangular system of polynomials within a given polybox in $\mathbb{C}^n$, for any given $\epsilon>0$. 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 numeric, certified and based on subdivision. We implemented it and compared it with two well-known homotopy solvers on various triangular systems. Our solver always gives correct answers, is often faster than the homotopy solver that often gives correct answers, and sometimes faster than the one that gives sometimes correct results.

cs.CG↗

arXiv2014

Numeric certified algorithm for the topology of resultant and discriminant curves

Let $\mathcal C$ be a real plane algebraic curve defined by the resultant of two polynomials (resp. by the discriminant of a polynomial). Geometrically such a curve is the projection of the intersection of the surfaces $P(x,y,z)=Q(x,y,z)=0$ (resp. $P(x,y,z)=\frac{\partial P}{\partial z}(x,y,z)=0$), and generically its singularities are nodes (resp. nodes and ordinary cusps). State-of-the-art numerical algorithms compute the topology of smooth curves but usually fail to certify the topology of singular ones. The main challenge is to find practical numerical criteria that guarantee the existence and the uniqueness of a singularity inside a given box $B$, while ensuring that $B$ does not contain any closed loop of $\mathcal{C}$. We solve this problem by first providing a square deflation system, based on subresultants, that can be used to certify numerically whether $B$ contains a unique singularity $p$ or not. Then we introduce a numeric adaptive separation criterion based on interval arithmetic to ensure that the topology of $\mathcal C$ in $B$ is homeomorphic to the local topology at $p$. Our algorithms are implemented and experiments show their efficiency compared to state-of-the-art symbolic or homotopic methods.

cs.CG↗

arXiv2014

Improved algorithm for computing separating linear forms for bivariate systems

We address the problem of computing a linear separating form of a system of two bivariate polynomials with integer coefficients, that is a linear combination of the variables that takes different values when evaluated at the distinct solutions of the system. The computation of such linear forms is at the core of most algorithms that solve algebraic systems by computing rational parameterizations of the solutions and this is the bottleneck of these algorithms in terms of worst-case bit complexity. We present for this problem a new algorithm of worst-case bit complexity $\sOB(d^7+d^6τ)$ where $d$ and $τ$ denote respectively the maximum degree and bitsize of the input (and where $\sO$ refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity). This algorithm simplifies and decreases by a factor $d$ the worst-case bit complexity presented for this problem by Bouzidi et al. \cite{bouzidiJSC2014a}. This algorithm also yields, for this problem, a probabilistic Las-Vegas algorithm of expected bit complexity $\sOB(d^5+d^4τ)$.

cs.CG↗

arXiv2014

Separating linear forms for bivariate systems

We present an algorithm for computing a separating linear form of a system of bivariate polynomials with integer coefficients, that is a linear combination of the variables that takes different values when evaluated at distinct (complex) solutions of the system. In other words, a separating linear form defines a shear of the coordinate system that sends the algebraic system in generic position, in the sense that no two distinct solutions are vertically aligned. The computation of such linear forms is at the core of most algorithms that solve algebraic systems by computing rational parameterizations of the solutions and, moreover, the computation a separating linear form is the bottleneck of these algorithms, in terms of worst-case bit complexity. Given two bivariate polynomials of total degree at most $d$ with integer coefficients of bitsize at most~$τ$, our algorithm computes a separating linear form in $\sOB(d^{8}+d^7τ)$ bit operations in the worst case, where the previously known best bit complexity for this problem was $\sOB(d^{10}+d^9τ)$ (where $\sO$ refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity).

cs.SC↗

arXiv2013

Rational Univariate Representations of Bivariate Systems and Applications

We address the problem of solving systems of two bivariate polynomials of total degree at most $d$ with integer coefficients of maximum bitsize $τ$. It is known that a linear separating form, that is a linear combination of the variables that takes different values at distinct solutions of the system, can be computed in $\sOB(d^{8}+d^7τ)$ bit operations (where $O_B$ refers to bit complexities and $\sO$ to complexities where polylogarithmic factors are omitted) and we focus here on the computation of a Rational Univariate Representation (RUR) given a linear separating form. We present an algorithm for computing a RUR with worst-case bit complexity in $\sOB(d^7+d^6τ)$ and bound the bitsize of its coefficients by $\sO(d^2+dτ)$. We show in addition that isolating boxes of the solutions of the system can be computed from the RUR with $\sOB(d^{8}+d^7τ)$ bit operations. Finally, we show how a RUR can be used to evaluate the sign of a bivariate polynomial (of degree at most $d$ and bitsize at most $τ$) at one real solution of the system in $\sOB(d^{8}+d^7τ)$ bit operations and at all the $Θ(d^2)$ {real} solutions in only $O(d)$ times that for one solution.

cs.SC↗