Making Waves

TL;DR

This paper investigates wave solutions to linear PDE constraints for vector-valued functions, exploring their geometric structure through projective varieties and syzygies, and generalizing classical wave cones.

Contribution

It introduces a geometric framework for classifying wave solutions using projective varieties, including determinantal and Fano varieties, extending the concept of wave cones.

Findings

Wave solutions correspond to distributions supported on low-dimensional varieties.

All waves can be parametrized by projective varieties derived from PDE support.

Special waves from vector potentials are represented by syzygies.

Abstract

We study linear PDE constraints for vector-valued functions and distributions. Our focus lies on wave solutions. These give rise to distributions with low-dimensional support. Special waves from vector potentials are represented by syzygies. We parametrize all waves by projective varieties derived from the support of the PDE. These include determinantal varieties and Fano varieties, and they generalize wave cones in analysis.