### Determinantal Representations and the Image of the Principal Minor Map

- Series
- Algebra Student Seminar
- Time
- Friday, December 9, 2022 - 10:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Abeer Al Ahmadieh – Georgia Institute of Technology – aahmadieh3@gatech.edu

The principal minor map takes an n by n square matrix to the length 2^n-vector of its principal minors. A basic question is to give necessary and sufficient conditions that characterize the image of various spaces of matrices under this map. In this talk, I will describe the image of the space of complex matrices using a characterization of determinantal representations of multiaffine polynomials, based on the factorization of their Rayleigh differences. Using these techniques I will give equations and inequalities characterizing the images of the spaces of real and complex symmetric and Hermitian matrices. For complex symmetric matrices, this recovers a result of Oeding from 2011. If time permits, I will also give examples to prove that for general matrices no such finite characterization is possible. This is based on joint work with Cynthia Vinzant.