On the computation of the topology of plane curves

International audienceLet P be a square free bivariate polynomial of degree at most d and with integer coefficients of bit size at most t. We give a deterministic algorithm for the computation of the topology of the real algebraic curve definit by P, i.e. a straight-line planar graph isotopic to the curve. Our main result is an algorithm for the computation of the local topology in a neighbourhood of each of the singular and critical points of the projection wrt the X axis in $\tilde{O} (d^6 t)$ bit operations where $\tilde{O}$ means that we ignore logarithmic factors in $d$ and $t$. Combined to state of the art sub-algorithms used for computing a Cylindrical Algebraic Decomposition, this result avoids a generic shear and gives a deterministic algorithm for the computation of the topology of the curve in $\tilde{O} (d^6 t + d^7)$ bit operations

Univariate real root isolation over a single logarithmic extension of real algebraic numbers

International audienceWe present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial B ∈ L[x], where L = Q[lg(α)] and α is a positive real algebraic number. The algorithm approximates the coefficients of B up to a sufficient accuracy and then solves the approximate polynomial. For this we derive worst case (aggregate) separation bounds. We also estimate the expected number of real roots when we draw the coefficients from a specific distribution and illustrate our results experimentally. A generalization to bivariate polynomial systems is also presented. We implemented the algorithm in C as part of the core library of mathematica for the case B ∈ Z[lg(q)][x] where q is positive rational number and we demonstrate its efficiency over various data sets

Univariate Real Root Isolation in Multiple Extension Fields

We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in Bα ∈ L[y], where L = (Q(α1,..., αℓ) is an algebraic extension of the rational numbers. Our bounds are single exponential in ℓ and match the ones presented in [34] for the case ℓ = 1. We consider two approaches. The first, indirect approach, using multivariate resultants, computes a univariate polynomial with integer coefficients, among the real roots of which are the real roots of Bα. The Boolean complexity of this approach is ÕB(N 4ℓ+4), where N is the maximum of the degrees and the coefficient bitsize of the involved polynomials. The second, direct approach, tries to solve the polynomial directly, without reducing the problem to a univariate one. We present an algorithm that generalizes Sturm algorithm from the univariate case, and modified versions of well known solvers that are either numerical or based on Descartes ’ rule of sign. We achieve a Boolean complexity of ÕB(min{N 4ℓ+7, N 2ℓ2 +6}) and ÕB(N 2ℓ+4), respectively. We implemented the algorithms in C as part of the core library of mathematica and we illustrate their efficiency over various data sets