Mixed determinants and the Kadison–Singer problem

Abstract

We adapt the arguments of Marcus, Spielman and Srivastava in their proof of the Kadison–Singer problem to prove improved paving estimates. Working with Anderson’s paving formulation of Kadison–Singer instead of Weaver’s vector balancing version, we show that the machinery of interlacing polynomials due to Marcus, Spielman and Srivastava works in this setting as well. The relevant expected characteristic polynomials turn out to be related to the so called “mixed determinants” that have been previously studied in a different context by Borcea and Branden. This approach allows us to show that any projection with diagonal entries 1/2 can be 4 paved, yielding improvements over the best known current estimates of 12. This approach also allows us to show that any projection with diagonal entries strictly less than 1/4 can be 2 paved, matching recent results of Bownik, Casazza, Marcus and Speegle. We also relate the problem of finding optimal paving estimates to bounding the root intervals of a natural one parameter deformation of the characteristic polynomial of a matrix that turns out to have several pleasing combinatorial properties.