computing with integrals in nonlinear algebra

March 9–25, 2021

A 6-lecture course virtually hosted by MPI-MiS 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.


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.


lecture #1, tue 9 mar 2021, 15h-17h (UTC+1)

Introduction. Differentially finite functions. Diffential equations as datastructure. Diagonals of rational functions.

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lecture #2, thu 11 mar 2021, 15h-17h (UTC+1)

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Local study of D-finite functions. Monodromy. Numerical evaluation of differentially finite functions. Computation of partial Taylor sums. Binary splitting.

lecture #3, tue 16 mar 2021, 15h-17h (UTC+1)

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Diagonals, constant terms and residues of rational functions. Binomial sums. D-finitess for these functions. Arithmetic properties.

lecture #4, thu 18 mar 2021, 15h-17h (UTC+1)

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Periods in experimental mathematics. Computation of the Picard group of a quartic surface.

lecture #5, tue 23 mar 2021, 15h-17h (UTC+1)

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Volume of semialgebraic sets. Method of moments.

lecture #6, thu 25 mar 2021, 15h-17h (UTC+1)

Problem solving. Please have Sagemath and ore_algebra installed.


A selection of articles that roughly define the scope of this lecture.


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

Numerical evaluation of differentially finite functions

Periods of rational integrals

Period numbers of curves and surfaces

Computation of volume of semialgebraic sets

Research directions