computing with integrals in nonlinear algebra
March 9–25, 2021
A 6lecture course virtually hosted by MPIMiS in Leipzig and held on zoom
Interested participants should register with Pierre Lairez or Bernd Sturmfels.
The lecture is over, but the material stays here.
abstract

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.
schedule
lecture #1, tue 9 mar 2021, 15h17h (UTC+1)
Introduction. Differentially finite functions. Diffential equations as datastructure. Diagonals of rational functions.
lecture #2, thu 11 mar 2021, 15h17h (UTC+1)
Local study of Dfinite functions. Monodromy. Numerical evaluation of differentially finite functions. Computation of partial Taylor sums. Binary splitting.
lecture #3, tue 16 mar 2021, 15h17h (UTC+1)
Diagonals, constant terms and residues of rational functions. Binomial sums. Dfinitess for these functions. Arithmetic properties.
lecture #4, thu 18 mar 2021, 15h17h (UTC+1)
Periods in experimental mathematics. Computation of the Picard group of a quartic surface.
lecture #5, tue 23 mar 2021, 15h17h (UTC+1)
Volume of semialgebraic sets. Method of moments.
lecture #6, thu 25 mar 2021, 15h17h (UTC+1)
Problem solving. Please have Sagemath and ore_algebra
installed.
references
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 semialgebraic 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 multipleprecision evaluation of Dfinite functions in Sagemath’.

Michałek, M. and Sturmfels, B. (2021), Invitation to nonlinear algebra

Molin, P. and Neurohr, C. (2019) ‘Computing period matrices and the AbelJacobi 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. ‘DModules and Holonomic Functions’
content
This is the plan. I will probably skip some subsections here and there, depending on the audience’s interests.
Linear differential equations as a data structure
 differentially finite (Dfinite) power series
 equivalence with recurrence relation
 example of Dfinite functions
 algebraic functions are Dfinite
 examples from combinatorics
 elementary closure properties
 addition, multiplication, some forms of composition
 solutions to linear ODEs
 ordinary points
 singular points
 exponents
 regular singularities
 formal solutions
 proof of identity
 guessing
 software packages
 gfun (Maple)
 ore_algebra (Sagemath)
 integration of multivariate Dfinite functions
Numerical evaluation of differentially finite functions
 radius of convergence of a Dfinite power series
 computation of partial Taylor sums
 naive quadratic algorithm
 binary splitting
 numerical analytic continuation
 decomposition in elementary steps
 regular singular points
 rigorous error bounds
 tail bounds
 fixed point method
 example: computing π
Periods of rational integrals
 definition
 relation with homology and cohomology
 topological reduction of periods
 analytic reduction of periods
 period matrix
 periods depending on a parameter…
 … are Dfinite (PicardFuchs equations)
 arithmetic properties
 BombieriDwork conjecture
 examples
 formal periods and diagonals
 definition
 they satisfy a PicardFuchs equations
 further arithmetic properties
 Furtsenberg theorem
 Christol conjecture
 binomial sums
 integral representations
 application to symbolic summation
 computation of PicardFuchs equations
 regular case
 singular case
Period numbers of curves and surfaces
 periods of curves
 period matrix
 Torelli’s theorem
 computation
 periods of surfaces
 period matrix
 Torellitype theorems
 computation
 effective Lefschetz (1,1)theorem
 LLL
 separation of periods of quartic surfaces
 higher dimensions
 Hodge conjecture
Computation of volume of semialgebraic sets
 statement of the problem
 function «volume of a slice»
 it is a period of rational function!
 everything in this course applies!
 an actual algorithm
 computing volume using moments
 computing moments using PicardFuchs equations
 further thoughts (WIP)
Research directions
 faster algorithms for computing periods
 revisiting the holonomic Dmodule approach to integration