TATERS

 Strength is a fundamental invariant measuring non-degeneracy for polynomials in many variables. First applied to asymptotically counting integer solutions of polynomial equations via the Hardy-Littlewood circle method, strength has more recently been used in additive combinatorics for studying distribution of polynomials over finite fields. We prove that, over many fields, the strength of a polynomial is equivalent to the codimension of its singular locus. This substantially simplifies the proofs of various known results and partially resolves a conjecture of Adiprasito, Kazhdan and Ziegler. No prior knowledge of the Hardy-Littlewood circle method will be assumed. Based on joint work with Benjamin Baily.

Mirroring Herb Wilf's much-cherished and still-wide-open question which polynomials are chromatic polynomials?, we give a brief survey of both the classification problem (the afore-mentioned question) and the detection problem (does a chromatic polynomial determine the graph?) for chromatic polynomials, which enumerate proper colorings of a given graph in terms of the number of colors. Inspired in part by the failure of the detection problem, Stanley introduced in the 1990s chromatic symmetric functions, which in turn opened up various lines of research. We introduce and study a q-version of the chromatic polynomial of a given graph G, defined as the sum of q^{l*c(v)} where l is a fixed integral linear form and the sum is over all proper n-colorings c of G. This turns out to be a polynomial in the q-integer [n]_q, with coefficients that are rational functions in q, and can be seen as an amphibian between chromatic polynomials and chromatic symmetric functions. We will exhibit several other structural results for q-chromatic polynomials and offer a strengthened version of Stanley's conjecture that the chromatic symmetric function distinguishes trees.

This talk is based on joint projects with Esme Bajo and Andrés Vindas-Meléndez.