Where are the roots of approximate polynomials? By combining the OM algorithm, cluster pictures, Berkovich skeletons and Brink’s continuity of roots inequality, we produce balls enclosing the roots of all the approximations at a given precision of a given univariate polynomial over a complete field with discrete valuation. Those balls can be presented in an approximate cluster picture or Berkovich skeleton. We present and showcase an algorithm to compute them.

In this paper, we study the connection between the OM-factorization of a polynomial and its cluster pictures, which is a representation of the relative configuration of the roots. Our contribution is threefold, assuming that the residual characteristic is zero or large enough (i.e. the field extension is tame) :(1)We provide and showcase an implementation of the OM algorithms.(2)We make explicit and constructive the connection between the valuative tree of a polynomial, the cluster picture of its roots and the Berkovich skeleton of its roots. As such, we provide a complexity result on the computation of cluster pictures.(3)We elaborate on this connection to provide and showcase an algorithm to compute cluster pictures based on the OM algorithms.

Twenty years ago, Zeng [28, 30] proposed floating-point algorithms to compute multiple roots of univariate polynomials with real or complex coefficients beyond the so-called “attainable accuracy barrier”. Based on these foundations, we propose a validated numeric point of view on this problem. Our first contribution is an improvement of Zeng’s multiplicity detection algorithm using a simple trick that allows us to recover much higher multiplicities. As our main contribution, we propose two floating-point validated algorithms to compute rigorous enclosures for multiple roots. They consist in carefully combining the ideas underlying Zeng’s numerical algorithms with Newton-like fixed-point validation techniques. We also provide a prototype Julia implementation of these algorithms.

We consider the problem of answering connectivity queries on a real algebraic curve. The curve is given as the real trace of an algebraic curve, assumed to be in generic position, and being defined by some rational parametrizations. The query points are given by a zero-dimensional parametrization.

We improve significantly the Nart-Montes algorithm for factoring polynomials over a complete discrete valuation ring A. Our first contribution is to extend the Hensel lemma in the context of generalised Newton polygons, from which we derive a new divide and conquer strategy. Also, if A has residual characteristic zero or high enough, we prove that approximate roots are convenient representatives of types, leading finally to an almost optimal complexity both for irreducibility and factorisation issues, plus the cost of factorisations above the residue field. For instance, to compute an OM-factorisation of F∈A[x], we improve the complexity results of [3] by a factor δ, the discriminant valuation of F.

Let K be a field of characteristic p with q elements and FΕ in L[X,Y] be a polynomial with p> deg_Y(F) and total degree d. In [40], we showed that rational Puiseux series of F above X=0 could be computed with an expected number of O~(d5+d3log q) arithmetic operations in L. In this paper, we reduce this bound to O~(d4+d2log q) using Hensel lifting and changes of variables in the Newton-Puiseux algorithm that give a better control of the number of steps. The only asymptotically fast algorithm required is polynomial multiplication over finite fields. This approach also allows to test the irreducibility of F in L[[X]][Y] with O(d3) operations in K. Finally, we describe a method based on structured bivariate multiplication [34] that may speed up computations for some input.

We present a method to approximate the signed distance function of a smooth curve or surface by using polynomial splines over hierarchical T-meshes (PHT splines). In particular, we focus on closed parametric curves in the plane and implicitly defined surfaces in space.

In [12], we sketched a numeric-symbolic method to compute Puiseux series with floating point coefficients. In this paper, we address the symbolic part of our algorithm. We study the reduction of Puiseux series coefficients modulo a prime ideal and prove a good reduction criterion sufficient to preserve the required information, namely Newton polygon trees. We introduce a convenient modification of Newton polygons that greatly simplifies proofs and statements of our results. Finally, we improve complexity bounds for Puiseux series calculations over finite fields, and estimate the bit-complexity of polygon tree computation.