On Singularity Properties of Word Maps and Applications to Probabilistic Waring Type Problems

Itay Glazer, Northwestern University, Evanston, Illinois, and Yotam I. Hendel, Northwestern University, Evanston, Illinois.

1. Introduction 2. Preliminaries 3. Some properties of the algebraic convolution operation 4. Lie algebra word maps 5. Lie algebra word maps: Proof of the main theorems 6. The commutator map revisited 7. Local behavior of word maps and a lower bound on the log canonical threshold of their fibers 8. Number theoretic interpretation of the flatness, $\varepsilon $-jet flatness and (FRS) properties 9. Applications to the $p$-adic probabilistic Waring type problem