A Productization Property/Trick For H-Stable(And Hopefully Strongly Log-Concave) Polynomials

          @misc{ scitalks_18805,
            doi = {},
            url = {https://simons.berkeley.edu/talks/productization-propertytrick-h-stableand-hopefully-strongly-log-concave-polynomials},
            author = {},
            keywords = {},
            language = {en},
            title = {A Productization Property/Trick For H-Stable(And Hopefully Strongly Log-Concave) Polynomials},
            publisher = {The Simons Institute for the Theory of Computing},
            year = {2021},
            month = {nov},
            note = {18805 see, \url{https://scitalks.ca/Simons-Institute/18805}}
          }
          

Abstract

The simplest homogeneous polynomials with nonnegative coefficients are products of linear forms Prod_{A}(X) associated with nonnegative matrices A. We prove that for any H-Stable(homogeneous and stable) polynomial p with P(E) = 1, where E is the vector of all ones, it's value p(X) = Prod_{A(X)}(X), where A(X) is nonnegative matrix with unit row sums and the vector of column sums equal to the gradient of p at E. I will first explain some intuition, and history, behind the result; sketch the proof and present a few applications and generalizations of this "productization" property. (Joint work with Jonathan Leake).