Computing periods of rational integrals

A period of a rational integral is the result of integrating, with respect to one or several variables, a rational function over a closed path. This work focuses particularly on periods depending on a parameter: in this case the period under consideration satisfies a linear differential equation, the Picard-Fuchs equation. I give a reduction algorithm that extends the Griffiths-Dwork reduction and apply it to the computation of Picard-Fuchs equations. The resulting algorithm is elementary and has been successfully applied to problems that were previously out of reach.Comment: To appear in Math. comp. Supplementary material at http://pierre.lairez.fr/supp/periods

The ideal membership problem and polynomial identity testing

AbstractGiven a monomial ideal I=〈m1,m2,…,mk〉 where mi are monomials and a polynomial f by an arithmetic circuit, the Ideal Membership Problem is to test if f∈I. We study this problem and show the following results.(a)When the ideal I=〈m1,m2,…,mk〉 for a constant k, we can test whether f∈I in randomized polynomial time. This result holds even for f given by a black-box, when f is of small degree.(b)When I=〈m1,m2,…,mk〉 for a constant kandf is computed by a ΣΠΣ circuit with output gate of bounded fanin, we can test whether f∈I in deterministic polynomial time. This generalizes the Kayal–Saxena result [11] of deterministic polynomial-time identity testing for ΣΠΣ circuits with bounded fanin output gate.(c)When k is not constant the problem is coNP-hard. We also show that the problem is upper bounded by coMAPP over the field of rationals, and by coNPModpP over finite fields.(d)Finally, we discuss identity testing for certain restricted depth 4 arithmetic circuits.For ideals I=〈f1,…,fℓ〉 where each fi∈F[x1,…,xk] is an arbitrary polynomial but k is a constant, we show similar results as (a) and (b) above

The Design and Implementation of a High-Performance Polynomial System Solver

This thesis examines the algorithmic and practical challenges of solving systems of polynomial equations. We discuss the design and implementation of triangular decomposition to solve polynomials systems exactly by means of symbolic computation. Incremental triangular decomposition solves one equation from the input list of polynomials at a time. Each step may produce several different components (points, curves, surfaces, etc.) of the solution set. Independent components imply that the solving process may proceed on each component concurrently. This so-called component-level parallelism is a theoretical and practical challenge characterized by irregular parallelism. Parallelism is not an algorithmic property but rather a geometrical property of the particular input system’s solution set. Despite these challenges, we have effectively applied parallel computing to triangular decomposition through the layering and cooperation of many parallel code regions. This parallel computing is supported by our generic object-oriented framework based on the dynamic multithreading paradigm. Meanwhile, the required polynomial algebra is sup- ported by an object-oriented framework for algebraic types which allows type safety and mathematical correctness to be determined at compile-time. Our software is implemented in C/C++ and have extensively tested the implementation for correctness and performance on over 3000 polynomial systems that have arisen in practice. The parallel framework has been re-used in the implementation of Hensel factorization as a parallel pipeline to compute roots of a polynomial with multivariate power series coefficients. Hensel factorization is one step toward computing the non-trivial limit points of quasi-components

Topologically certified approximation of umbilics and ridges on polynomial parametric surface

Given a smooth surface, a blue (red) ridge is a curve along which the maximum (minimum) principal curvature has an extremum along its curvature line. Ridges are curves of extremal curvature and encode important informations used in surface analysis or segmentation. But reporting the ridges of a surface requires manipulating third and fourth order derivatives whence numerical difficulties. Additionally, ridges have self-intersections and complex interactions with the umbilics of the surface whence topological difficulties. In this context, we make two contributions for the computation of ridges of polynomial parametric surfaces. First, by instantiating to the polynomial setting a global structure theorem of ridge curves proved in a companion paper, we develop the first certified algorithm to produce a topological approximation of the curve P encoding all the ridges of the surface. The algorithm exploits the singular structure of P umbilics and purple points, and reduces the problem to solving zero dimensional systems using Gröbner basis. Second, for cases where the zero-dimensional systems cannot be practically solved, we develop a certified plot algorithm at any fixed resolution. These contributions are respectively illustrated for Bezier surfaces of degree four and five