December 12,  2025

on-line on zoom

• 10:00-10:50 Tomasz Kowalczyk (Uniwersytet Jagielloński) On Hilbert's 17th problem

  Abstract: In its original statement, Hilbert's 17th problem is the problem whether a nonnegative polynomial on \(\mathbb{R}^n\) is a sum of squares of rational functions. Given a ring of real valued functions on some topological space \(X\) we can ask a similar question: which functions can be written as a sum of squares in the total ring of fractions of \(R\), and if so, how many squares are needed? We will be mostly interested in the second part, namely computation of the Pythagoras numbers. For a commutative ring \(R\) with identity, we define \(p_{2d}(R)\), \(2d\)-Pythagoras number of \(R\), as the smallest positive integer \(g\) such that any sum of \(2d\)th powers can be written as a sum of at most \(g\) \(2d\)th powers. If such number does not exist, we define \(p_{2d}(R)=\infty\). During the talk, I will present both classical and new results concerning sums of even powers in rings of polynomials over a formally real field, rings of \(k\)-regulous functions, rings of regular functions on a nonsingular rational surface. Also, the notion of a bad set of a nonnegative function will be discussed. Presentation will be based on:
  • Banecki J., Kowalczyk T., Sums of even powers of k-regulous functions, Indag. Math., New Ser., 34(3):477–487, (2023).
  • Kowalczyk T., Sums of squares of regular functions on rational surfaces, arXiv:2407.20378.
  • Kowalczyk T., Vill J.,On higher Pythagoras numbers of real polynomial rings, Indiana Univ. Math. J., (2025), to appear.

•  11:00-11:50 Thibault Chailleux (Universite d'Angers) A global kinematic formula for definable closed sets

  Abstract:  We will begin by briefly introducing the notion of kinematic formula through the classical Cauchy-Crofton formula that relates the length of a rectifiable curve to the integral over all affine lines of the number of intersection points with the line. Similarly, we will present the global kinematic formula in the case of closed definable sets as a formula linking a curvature invariant of a first set to the Euler characteristic of its intersection with a second set, integrating this last quantity over all affine transformations of the second set. We will also compare this formula to the infinitesimal kinematic formula for germs of closed definable sets, and attempt to illustrate the differences in the methods used to prove the global formula in comparison to the local one.