We address univariate root isolation when the polynomial’s coefficients are in a multiple field extension. We consider a polynomial F ∈ L[Y], where L is a multiple algebraic extension of . We provide aggregate bounds for F and algorithmic and bit-complexity results for the problem of isolating its roots.

We consider the problem of computing the topology and describing the geometry of a parametric curve in Rn. We present an algorithm, PTOPO, that constructs an abstract graph that is isotopic to the curve in the embedding space. Our method exploits the benefits of the parametric representation and does not resort to implicitization.

Very recent work introduces an asymptotically fast subdivision algorithm, denoted ANewDsc, for isolating the real roots of a univariate real polynomial. The method combines Descartes? Rule of Signs to test intervals for the existence of roots, Newton iteration to speed up convergence against clusters of roots, and approximate computation to decrease the required precision. It achieves record bounds on the worst-case complexity for the considered problem, matching the complexity of Pan's method for computing all complex roots and improving upon the complexity of other subdivision methods by several magnitudes. In the article at hand, we report on an implementation of ANewDsc on top of the RS root isolator. RS is a highly efficient realization of the classical Descartes method and currently serves as the default real root solver in Maple. We describe crucial design changes within ANewDsc and RS that led to a high-performance implementation without harming the theoretical complexity of the underlying algorithm.

We propose a set of simple and fast algorithms for evaluating and using trigonometric expressions in the form F=Σkd=0fkΕZd‹ n, for a fixed nin>0:ΕZ > 0: computing the sign of such an expression, evaluating it numerically and computing its minimal polynomial in Q[x]. As critical byproducts, we propose simple and efficient algorithms for performing arithmetic operations (multiplication, division, gcd) on polynomials expressed in a Chebyshev basis (with the same bit-complexity as in the monomial basis) and for computing the minimal polynomial of 2 cos π over n in Õ(n02) bit operations with n0 ≤ n is the odd squarefree part of n. Within such a framework, we can decide if F=0 in Õ(d(τ+d)) bit operations, compute the sign of F in Õ(d2τ) bit operations and compute the minimal polynomial of F in Õ(n3τ) bit operations, where τ denotes the maximum bitsize of the f k's.

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 ÕB(d7 + d6τ) where d and τ denote respectively the maximum degree and bitsize of the input (and where Õ refers to the complexity where polylogarithmic factors are omitted and OB 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. [5]. This algorithm also yields, for this problem, a probabilistic Las-Vegas algorithm of expected bit complexity ÕB(d5 + d4τ).

Let P ∈ Z[X, Y] be a square-free polynomial and C(P):= {(α, β) ∈ R2, P(α, β) = 0} be the real algebraic curve defined by P. Our main result is an algorithm for the computation of the local topology in a neighbourhood of each of the singular points and critical points of the projection wrt the X-axis in Õ(d6τ+d7) bit operations where Õ means that we ignore logarithmic factors in d and τ. Compared 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 C(P) i.e a straight-line planar graph isotopic to C(P) in Õ(d6τ + d7) bit operations.

We address the problem of solving systems of two bivariate polynomials of total degree at most d with integer coefficients of maximum bitsize τ We suppose known a linear separating form (that is a linear combination of the variables that takes different values at distinct solutions of the system) and focus on the computation of a Rational Univariate Representation (RUR).

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 of a separating linear form is the bottleneck of these algorithms, in terms of worst-case bit complexity.

We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position.

Classifying the Perspective-Three-Point problem (abbreviated by P3P in the sequel) consists in determining the number of possible positions of a camera with respect to the apparent position of three points. In the case where the three points form an isosceles triangle, we give a full classification of the P3P. This leads to consider a polynomial system of polynomial equations and inequalities with 4 parameters which is generically zero-dimensional. In the present situation, the parameters represent the apparent position of the three points so that solving the problem means determining all the possible numbers of real solutions with respect to the parameters' values and give a sample point for each of these possible numbers. One way for solving such systems consists first in computing a discriminant variety. Then, one has to compute at least one point in each connected component of its real complementary in the parameter's space. The last step consists in specializing the parameters appearing in the initial system by these sample points. Many computational tools may be used for implementing such a general method, starting with the well known Cylindrical Algebraic Decomposition (CAD in short), which provides more information than required. In a first stage, we propose a full algorithm based on the straightforward use of some sophisticated software such as FGb (Grobner bases computations) RS (real roots of zero-dimensional systems), DV (Discriminant varieties) and RAGlib (Critical point methods for semi-algebraic systems). We then improve the global algorithm by refining the required computable mathematical objects and related algorithms and finally provide the classification. Three full days of computation were necessary to get this classification which is obtained from more than 40000 points in the parameter's space.