# Alain Yger: Ronkin’s and Crofton’s formulae, as I learned from Mikaël

Mikael Passare Day

**Time: **
Wed 2021-09-15 13.15 - 14.00

**Location: **
Stockholm University, Kräftriket, Room 14 and Zoom 694 2604 1151

**Participating: **
Alain Yger (IMB, Bordeaux University)

### Abstract

Mikaël’s so clever intuitions remained a permanent source of inspiration for more than thirty years. The purpose of my talk is to convince you they still do. I shall focus in this talk about two concepts, among so many, I got familar through him. The first one is that of *Ronkin convex function* linked with a Laurent polynomial *F* in *n* complex variables, together with its companions, each of them being attached to the *p*-adic absolute value (in view of the product formula) in case *F* has rational coefficients. The second one is *Crofton’s averaging formula* (inherent to the Fubini–Study metric on the projective space), which connects multiplicative residue calculus (interpreted in terms of currents, as Mikaël learnt us how to do, in order to profit from analysis flexibility) with Bochner–Martinelli residue calculus. It appears that Ronkin function, as Mahler measure does in the projective space, is, once paired with its ultrametric companions, a marker for the logarithmic height (that is in fact the arithmetic complexity) of the algebraic hypersurface defined by *F* in the affine *n* dimensional complex torus. On the other hand, what one could call *croftonisation* makes a bridge from closed parametric formulae for Hilbert’s nullstellensatz over \(\mathbb{Q}\) to explicit realizations (precisely through weighted Bochner–Martinelli integral formulae), due to Mats Andersson and Elin Götmark, of Briançon-Skoda theorem in \(\mathbb{C}[X_1,\dots,X_n]\); it leads also to the concept of *generalized cycle (or its class)*, which archetypical examples are Segre or Stückrad–Vogel generalized cycles or their classes. My talk is based on the works of Mats Andersson, Dennis Eriksson, Håkan Samuelsson-Kalm, Elizabeth Wulcan, Martin Sombra, Alekos Vidras, Roberto Gualdi, Farhad Babaee, in which I was involved as collaborator (or, in case of Roberto and Farhad, as thesis director).