# Bounds for polynomials on algebraic numbers and application to curve topology

2 GAMBLE - Geometric Algorithms and Models Beyond the Linear and Euclidean realm
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : Let $P \in \mathbb{Z} [X, Y]$ be a given square-free polynomial of total degree $d$ with integer coefficients of bitsize less than $\tau$, and let $V_{\mathbb{R}} (P) := \{ (x,y) \in \mathbb{R}^2, P (x,y) = 0 \}$ be the real planar algebraic curve implicitly defined as the vanishing set of $P$. We give a deterministic and certified algorithm to compute the topology of $V_{\mathbb{R}} (P)$ in terms of a straight-line planar graph $\mathcal{G}$ that is isotopic to $V_{\mathbb{R}} (P)$. Our analysis yields the upper bound $\tilde O (d^5 \tau + d^6)$ on the bit complexity of our algorithm, which matches the current record bound for the problem of computing the topology of a planar algebraic curve However, compared to existing algorithms with comparable complexity, our method does not consider any change of coordinates, and the returned graph $\mathcal{G}$ yields the cylindrical algebraic decomposition information of the curve. Our result is based on two main ingredients: First, we derive amortized quantitative bounds on the the roots of polynomials with algebraic coefficients as well as adaptive methods for computing the roots of such polynomials that actually exploit this amortization. Our second ingredient is a novel approach for the computation of the local topology of the curve in a neighborhood of all critical points.
Document type :
Preprints, Working Papers, ...

https://hal.inria.fr/hal-01891417
Contributor : Fabrice Rouillier <>
Submitted on : Tuesday, October 9, 2018 - 3:31:21 PM
Last modification on : Friday, November 8, 2019 - 11:00:36 AM

### Identifiers

• HAL Id : hal-01891417, version 1
• ARXIV : 1807.10622

### Citation

Daouda Niang Diatta, Sény Diatta, Fabrice Rouillier, Marie-Françoise Roy, Michael Sagraloff. Bounds for polynomials on algebraic numbers and application to curve topology. 2018. ⟨hal-01891417⟩

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