Optimal point configurations and Bombieri-type inequalities
Organizers
Speaker
Time
Wednesday, March 25, 2026 1:00 PM - 2:30 PM
Venue
A3-4-301
Online
Zoom 242 742 6089
(BIMSA)
Abstract
In recent work, Etayo introduces a new Bombieri-type inequality for monic polynomials. Here we reinterpret this new inequality as a more general integral inequality involving the Green function for the sphere. This rather geometric interpretation allows for generalizations of the basic inequality, involving fractional zeros while also opening up the possibility to extend the setting to general compact Riemann surfaces. We derive a sharp form of these generalized Bombieri-type inequalities for the case of the sphere and the torus. These inequalities involve a quantity we call the packing number, which in turn is inspired by the geometric zero packing problems considered by Hedenmalm in the context of the asymptotic variance of the Bergman projection of a bounded function. As for the torus, we introduce analogs of polynomials (pseudopolynomials) based on the classical Weierstrass \sigma function, and we explain how such pseudopolynomials fit in with the extended geometric Bombieri-type inequality. The sharpness of the packing number bound on the torus involves the construction of a lattice configuration on the torus for any given integer number of points. The corresponding bound for the sphere instead relies on the existence of well-conditioned polynomials in the sense of Shub and Smale.
Speaker Intro
Hedenmalm has mainly contributed to the development of the theory of Bergman spaces and the associated reproducing kernels in one complex variable. In 1996 he became a professor at Lund University and in 1997 he was elected to KFS, the Royal Physiographic Society in Lund. Later, in 2018, he was elected to DKNVS, the Royal Norwegian Society of Sciences and Letters in Trondheim. Hedenmalm has collaborated with a number of other mathematicians, in particular with Alexander Borichev, Serguei Shimorin and Nikolai Makarov. Since 2002 he is professor at the Royal Institute of Technology (KTH) in Stockholm. He received the Wallenberg Prize in 1992, and in 1996 he was invited speaker at 2ECM (second European Congress of Mathematicians) in Budapest. In 2000 he received the Göran Gustafsson Prize (KVA). In 2015, he received the Eva and Lars Gårding Prize from KFS, the Royal Physiographic Society in Lund.