A 6-lecture course virtually hosted by MPI-MiS in Leipzig and held on zoom
March 9–25, 2021
Interested participants should register with Pierre Lairez or Bernd Sturmfels.
How to compute 100 digits of the volume of a semialgebraic (defined by polynomial inequalities)?
How to compute the moments $\int_{[0,1]^n} f(x_1,\dotsc,x_n)^k \mathrm{d}x_1 \dotsc \mathrm{d}x_n$ for large $k$? This problem stems from polynomial optimization.
How to compute a recurrence relation for the numbers $\sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2$? This relation is central in Apéry's proof of the irrationality of $\zeta(3)$.
How to count the number of smooth rational curves of degree $d$ on a smooth quartic surface in $\mathbb{P}^3$?
These questions from diverse area of mathematics all feature period functions of rational integrals: analytic functions defined by integrating multivariate rational functions. In some regards, period functions are the simplest nonalgebraic functions. One of the first instance to be studied was the perimeter of the ellipse, as a function eccentricity.
I will first expose the fundamental material to compute with period function: linear differential equations as a data structure, symbolic integration and numerical analytic continuation. Next, I will show how to apply these technique in practice on many different problems, including the four questions above. As much as possible, I will connect with current research questions.
Introduction. Differentially finite functions. Diffential equations as datastructure. Diagonals of rational functions.
Numerical evaluation of differentially finite functions. Computation of partial Taylor sums. Rigorous error bounds.
A selection of articles that roughly define the scope of this lecture.
Bostan, A., Lairez, P. and Salvy, B. (2017) ‘Multiple binomial sums’, Journal of Symbolic Computation, 80, pp. 351–386.
Griffiths, P. A. (1969) ‘On the periods of certain rational integrals’, Annals of Mathematics, 90, pp. 460–541.
Hoeven, J. van der (2001) ‘Fast evaluation of holonomic functions near and in regular singularities’, Journal of Symbolic Computation, 31(6), pp. 717–743.
Lairez, P. (2016) ‘Computing periods of rational integrals’, Mathematics of Computation, 85(300), pp. 1719–1752.
Lairez, P., Mezzarobba, M. and Safey El Din, M. (2019) ‘Computing the volume of compact semi-algebraic sets’, ISSAC 2019.
Lairez, P. and Sertöz, E. C. (2019) ‘A numerical transcendental method in algebraic geometry’, SIAM Journal on Applied Algebra and Geometry, pp. 559–584.
Lasserre, J. B. (2019) ‘Volume of sublevel sets of homogeneous polynomials’, SIAM Journal on Applied Algebra and Geometry, 3(2), pp. 372–389.
Mezzarobba, M. (2016) ‘Rigorous multiple-precision evaluation of D-finite functions in Sagemath’.
Michałek, M. and Sturmfels, B. (2021), Invitation to nonlinear algebra
Molin, P. and Neurohr, C. (2019) ‘Computing period matrices and the Abel-Jacobi map of superelliptic curves’, Mathematics of Computation, 88(316), pp. 847–888.
Oaku, T. (2013) ‘Algorithms for integrals of holonomic functions over domains defined by polynomial inequalities’, Journal of Symbolic Computation, 50, pp. 1–27.
Salvy, B. (2019) ‘Linear differential equations as a data structure’, Foundations of Computational Mathematics, 19(5), pp. 1071–1112.
Sturmfels, B. and Sattelberger, A.-L. ‘D-Modules and Holonomic Functions’
This is the plan. I will probably skip some subsections here and there, depending on the audience's interests.