74 citations · 304 across the 21 of their papers we have counts for
21 papers
A semi-numerical algorithm for the homology lattice and periods of complex elliptic surfaces over the projective line
Eric Pichon-Pharabod
We provide an algorithm for computing an effective basis of homology of elliptic surfaces over the complex projective line on which integration of periods can be carried out. This…
Effective homology and periods of complex projective hypersurfaces
Pierre Lairez, Eric Pichon-Pharabod, Pierre Vanhove
We introduce a new algorithm for computing the periods of a smooth complex projective hypersurface. The algorithm intertwine with a new method for computing an explicit basis of th…
A Direttissimo Algorithm for Equidimensional Decomposition
Christian Eder, Pierre Lairez, Rafael Mohr +1
We describe a recursive algorithm that decomposes an algebraic set into locally closed equidimensional sets, i.e. sets which each have irreducible components of the same dimension.…
Axioms for a theory of signature bases
Pierre Lairez
Twenty years after the discovery of the F5 algorithm, Gröbner bases with signatures are still challenging to understand and to adapt to different settings. This contrasts with Buch…
Algorithms for minimal Picard-Fuchs operators of Feynman integrals
Pierre Lairez, Pierre Vanhove
In even space-time dimensions the multi-loop Feynman integrals are integrals of rational function in projective space. By using an algorithm that extends the Griffiths--Dwork reduc…
A Signature-based Algorithm for Computing the Nondegenerate Locus of a Polynomial System
Christian Eder, Pierre Lairez, Rafael Mohr +1
Polynomial system solving arises in many application areas to model non-linear geometric properties. In such settings, polynomial systems may come with degeneration which the end-u…
Computing the dimension of real algebraic sets
Piere Lairez, Mohab Safey El Din
Let be the set of real common solutions to in and be the maximum total degree of the 's. We design an algorithm…
Separation of periods of quartic surfaces
Pierre Lairez, Emre Can Sertöz
We give a computable lower bound on the distance between two distinct periods of a given quartic surface defined over the algebraic numbers. The main ingredient is the determinatio…
Rigid continuation paths II. Structured polynomial systems
Peter Bürgisser, Felipe Cucker, Pierre Lairez
This work studies the average complexity of solving structured polynomial systems that are characterized by a low evaluation cost, as opposed to the dense random model previously u…
Computing the volume of compact semi-algebraic sets
Pierre Lairez, Marc Mezzarobba, Mohab Safey El Din
Let be a compact basic semi-algebraic set defined as the real solution set of multivariate polynomial inequalities with rational coefficients. We design an algorithm…
A numerical transcendental method in algebraic geometry
Pierre Lairez, Emre Can Sertöz
Based on high precision computation of periods and lattice reduction techniques, we compute the Picard group of smooth surfaces. We also study the lattice reduction technique that…
Generalized Hermite Reduction, Creative Telescoping and Definite Integration of D-Finite Functions
Alin Bostan, Frédéric Chyzak, Pierre Lairez +1
Hermite reduction is a classical algorithmic tool in symbolic integration. It is used to decompose a given rational function as a sum of a function with simple poles and the deriva…
Rigid continuation paths I. Quasilinear average complexity for solving polynomial systems
Pierre Lairez
How many operations do we need on the average to compute an approximate root of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked whether a polynomial bou…
Computing the homology of basic semialgebraic sets in weak exponential time
Peter Bürgisser, Felipe Cucker, Pierre Lairez
We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets which works in weak exponential time. That is,…
On p-adic differential equations with separation of variables
Pierre Lairez, Tristan Vaccon
Several algorithms in computer algebra involve the computation of a power series solution of a given ordinary differential equation. Over finite fields, the problem is often lifted…
The boundary of the orbit of the 3 by 3 determinant polynomial
Jesko Hüttenhain, Pierre Lairez
We consider the 3 by 3 determinant polynomial and we describe the limit points of the set of all polynomials obtained from the determinant polynomial by linear change of variables.…
Computing the Chow variety of quadratic space curves
Peter Bürgisser, Kathlén Kohn, Pierre Lairez +1
Quadrics in the Grassmannian of lines in 3-space form a 19-dimensional projective space. We study the subvariety of coisotropic hypersurfaces. Following Gel'fand, Kapranov and Zele…
A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time
Pierre Lairez
We describe a deterministic algorithm that computes an approximate root of n complex polynomial equations in n unknowns in average polynomial time with respect to the size of the i…
Computing periods of rational integrals
Pierre Lairez
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 pe…
Creative telescoping for rational functions using the Griffiths-Dwork method
Alin Bostan, Pierre Lairez, Bruno Salvy
Creative telescoping algorithms compute linear differential equations satisfied by multiple integrals with parameters. We describe a precise and elementary algorithmic version of t…
Resolution except for minimal singularities II. The case of four variables
Edward Bierstone, Pierre Lairez, Pierre D. Milman
In this sequel to Resolution except for minimal singularities I, we find the smallest class of singularities in four variables with which we necessarily end up if we resolve singul…