News and agenda

  • March 27, 2015 conference

    Diagonals of rational functions, main conference of Jean Morlet chair: “Artin approximation and infinite dimensional geometry”. Slides

  • March 17, 2015 seminar

    Sommes binomiales multiples et diagonales de fractions rationnelles, séminaire tournant de théorie des nombres, Grenoble. Slides

  • February 04, 2015 conference

    Sommes binomiales multiples et diagonales de fractions rationnelles, journées combinatoires de Bordeaux. Slides

  • December 15, 2014 poster

    Poster presented at FoCM 2014. hal

  • November 12, 2014 breaking news

    PhD defense. Slides

  • November 03, 2014 conference

    Sommes binomiales multiples : structure et calcul, journées nationales de calcul formel. Slides

  • September 01, 2014 breaking news

    I move to Berlin.

  • May 29, 2014 seminar

    Binomial sums and integral representations, MAGIC seminar, Imperial College, Londres.

  • April 20, 2014 preprint

    “Computing periods of rational integrals”.

  • April 11, 2014 seminar

    Représentations intégrales des sommes binomiales, séminaire Teich, Marseille. Slides

  • February 14, 2014 conference

    Un nouvel algorithme pour calculer les périodes des intégrales rationnelles, journées holonomes, Grenoble. Slides

  • October 18, 2013 seminar

    Le calcul des intégrales multiples de fractions rationnelles : de Hermite à Dimca, Institut de recherche mathématique de Rennes. Slides

  • September 16, 2013 seminar

    Algorithmes pour le calcul des intégrales multiples, Laboratoire Dieudonné, Nice.

  • June 28, 2013 conference

    A polynomial time algorithm for rational creative telescoping, ISSAC 2013, Boston, USA. Slides

  • May 14, 2013 conference

    Création télescopique rationnelle en temps polynomial, journées nationales de calcul formel.

  • March 07, 2013 seminar

    Diagonales de fractions rationnelles et le calcul des périodes des intégrales multiples, séminaire des thésards de l’Institut de mathématiques de Jussieu, université Paris VI. Slides

  • February 19, 2013 seminar

    Calcul des périodes des intégrales multiples, séminaire de l’Institut C. Jordan, université Lyon I.

  • January 24, 2013 seminar

    Calcul des périodes des intégrales multiples, séminaire AriC, LIP, ENS Lyon.

  • January 18, 2013 preprint

    “Creative telescoping for rational functions using the Griffiths-Dwork method”.


September 2014 — Present
Wissenschaftlicher Mitarbeiter (i.e. post-doc) in the team of Peter Bürgisser. Logo TU Berlin

TU BerlinFachgebiet Algebra & Zahlentheorie

September 2011 – August 2014
PhD candidate. Logo Inria
Advisors: Bruno Salvy and Alin Bostan.

Inria SaclayÉquipe Specfun

September 2010 – May 2011
Visiting member in Fields Institute, Toronto. Logo Fields Institute

Advisor: Edward Bierstone.

Recent work

The Chow variety of quadratic space curves

August 2015

It is well known that lines of the projective space $\mathbb P^3$ — or equivalently, 2-dimensional subspaces of a 4-dimensional vector space — are parametrized by a projective variety of dimension four, the Grassmannian $G(2,4)$. And Plücker coordinates realize this variety as a quadric of $\mathbb P^5$. What about the parametrization curves of degree two in $\mathbb P^3$? Celebrated predecessors — Cayley, van der Waerden, Green, Morrison, Gel’fand, Kapranov, Zelevinsky, etc. — have shown how to realize this parametrization as a 8-dimensional subvariety of $\mathbb P^{19}$: the Chow variety of quadratic space curves.

In “Computing the Chow variety of quadratic space curves”, we performed the computations all the way through to give the explicit equations. For example, we obtain that the two components of the Chow variety of quadratic space curves have degree 140 and 92.

Deterministic algorithm to compute approximate roots of polynomial systems

July 2015

For numerically solving polynomial system of equation, the method of homotopy continuation is widely used. They are a lot of questions related to its computational complexity, the first of which is Smale's 17th problem. In a recent breakthrough, Berltrán and Pardo have shown how to achieve a polynomial average complexity with a probabilistic algorithm.

In “A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time”, I show how to turn this algorithm into a deterministic one. This gives an answer to Smale's 17th problem in the BSS model.


  • August 2015 conference paper

    Computing the Chow variety of quadratic space curves

    by Peter Bürgisser, Kathlén Kohn, Pierre Lairez and Bernd Sturmfels

    MACIS 2015, Applied algebraic geometry session, Berlin

    Abstract. 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 Zelevinsky, it decomposes into Chow forms of plane conics, Chow forms of pairs of lines, and Hurwitz forms of quadric surfaces. We compute the ideals of these loci.

    • August 29, 2015 (preprint)
  • July 2015 preprint

    A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time

    by Pierre Lairez

    Abstract. 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 input, in the Blum-Shub-Smale model with square root. It rests upon a derandomization of an algorithm of Beltrán and Pardo and gives a deterministic affirmative answer to Smale’s 17th problem. The main idea is to make use of the randomness contained in the input itself.

    • July 20, 2015 (preprint)
  • March 2015 journal paper

    Computing periods of rational integrals

    by Pierre Lairez

    Mathematics of computation, to appear

    Supplementary material

    Abstract. 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.

    • April 20, 2014 (preprint)
    • January 31, 2015 (revision)
    • March 13, 2015 (accepted)
  • November 2014 thesis

    Périodes d’intégrales rationnelles : algorithmes et applications

    by Pierre Lairez

    Thèse de doctorat, prix de thèse de l’École polytechnique.

    Abstract. A period of rational integral is the result of integrating, with respect to one or several variables, a rational function along a closed path. When the period under consideration depends on a parameter, it satisfies a specific linear differential equation called Picard-Fuchs equation. These equations and their computation are important for computer algebra, but also for algebraic geometry (they contains geometric invariants), in combinatorics (many generating functions are periods) or in theoretical physics. This thesis offers and studies algorithms to compute them.

    The first chapter shows bounds on the size of Picard-Fuchs equations and on the complexity of their computation. Existing algorithms for computing these equations often produce, in the same time, certificates, typically huge, which allows to check afterwards the correctness of the equation. The bounds I obtained enlighten the computational nature of Picard-Fuchs equations, they show in particular that the certificates are not a required byproduct. The proof relies on the study of the generic case and the reduction of pole order with Griffiths-Dwork method.

    The second chapter offers an algorithms for computing Picard-Fuchs equations more efficiently. It allows for the resolution of many previously unsolved problems. It relies on a method for reducing the pole order which extends Griffiths-Dwork reduction to the singulars cases.

    The third chapter draws a rigorous correspondence between periods of rational integrals and generating functions of multiple binomial sums. Together with the computation of Picard-Fuchs equations, it allows automatically proving identities about binomial sums.

    Jury de thèse :

    • November 12, 2014 (defense)
  • June 2013 conference paper

    Creative telescoping for rational functions using the Griffiths-Dwork method

    by Alin Bostan, Pierre Lairez and Bruno Salvy

    Proceedings of ISSAC 2013 (Boston, USA), distinguished student author award

    Abstract. Les algorithmes de création télescopique calculent les équations différentielles linéaires vérifiées par les intégrales multiples à paramètre. Nous décrivons, en utilisant la méthode de Griffiths-Dwork, un tel algorithme pour les fractions rationnelles. Cela permet d’obtenir des bornes sur l’ordre et le degré des coefficients des équations différentielles recherchées, ainsi qu’au premier résultat de complexité qui est simplement polynomial en le nombre de variables. Un aspect important de l’algorithme est que le calcul du certificat n’est pas nécessaire. L’intérêt pratique de l’approche est démontré par une première implémentation.

    • January 18, 2013 (preprint)
    • April 21, 2013 (revision)
    • March 22, 2013 (accepted)
    • June 6, 2013 (print)
  • December 2012 journal paper

    Resolution except for minimal singularities II. The case of four variables

    by Edward Bierstone, Pierre Lairez and Pierre Milman

    Advances in Mathematics, 2012, vol. 231, no. 5, pp. 3003–3021

    Abstract. 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 singularities except for normal crossings. The main new feature is a characterization of singularities in four variables which occur as limits of triple normal crossings singularities, and which cannot be eliminated by a birational morphism that avoids blowing up normal crossings singularities.

    • July 27, 2011 (preprint)
    • August 1, 2012 (accepted)
    • December 1, 2012 (print)




BinomSums is a package for the computer algebra system Maple that provides functions to handle multiple binomial sums through integral representations. It implements the algorithms described in my PhD thesis “Périodes d’intégrales rationnelles : algorithmes et applications”.

GitHub repo



Periods is a package for the computer algebra system Magma that provides functions to compute periods of rational integrals. It implements the algorithm described in “Computing periods of rational integrals”. Several computational premières have been performed with it and it has been critical in augmenting and the database of Calabi-Yau operators. It is also capable of integrating the integral representations that the package BinomSums outputs.

GitHub repo