Smoothness and Determinantal Representations of Adjoint Hypersurfaces

Date

Nov 20, 2025

Time

3:00 PM - 3:45 PM

Speaker

Clemens Brüser

Affiliation

TU Dresden

Language

en

Main Topic

Biologie

Host

Local Organisers: Nikola Sadovek, Maximilian Wiesmann, Giulio Zucal

Description

Adjoint polynomials of convex polytopes have recently received attention from the field of particle physics, and the question has been raised whether they admit determinantal representations. In this talk we define the notion of adjoint polynomials/hypersurfaces and characterize them through their degree and a simple vanishing condition. Through this vanishing condition we derive a certificate for the existence of singularities on the adjoint hypersurface. We then survey the classical theory on determinantal representations. We prove that the adjoint curve of a polygon always has a natural symmetric determinantal representation that certifies hyperbolicity. For three-dimensional polytopes we show that if the adjoint is smooth, then a determinantal representation exists. The methods to find these representations are computationally viable. There are also some negative results for higher dimensions. The presented results are based on joint work with Mario Kummer and Dmitrii Pavlov (both TU Dresden) and with Julian Weigert (MPI-MIS Leipzig).

Last modified: Nov 20, 2025, 7:36:20 AM