Bounds for Polynomials on Algebraic Numbers and Application to Curve Topology

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 $V_{{\mathbb {R}}} (P) := \{ (x,y) \in {\mathbb {R}}^2\mid P (x,y) = 0 \}$ be the real planar algebraic curve implicitly defined as the vanishing set of P. We give a deterministic algorithm to compute the topology of $V_{{\mathbb {R}}} (P)$ in terms of a simple straight-line planar graph ${\mathcal {G}}$ that is isotopic to $V_{{\mathbb {R}}} (P)$. The upper bound on the bit complexity of our algorithm is in ${\tilde{O}}(d^5 \tau + d^6)$(The expression “the complexity is in ${\tilde{O}}(f(d,\tau ))$” with f a polynomial in $d,\tau$ is an abbreviation for the expression “there exists a positive integer c such that the complexity is in $O(({\log d }\log \tau )^c f(d,\tau ))$”); 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 more importantly the returned simple planar graph ${\mathcal {G}}$ yields the cylindrical algebraic decomposition information of the curve in the original coordinates. Our result is based on two main ingredients: First, we derive amortized quantitative bounds on the roots of polynomials with algebraic coefficients as well as adaptive methods for computing the roots of bivariate polynomial systems that actually exploit this amortization. The results we obtain are more general that the previous literature. Our second ingredient is a novel approach for the computation of the local topology of the curve in a neighborhood of all singular points.