Title: Higher Order Arithmetic-Geometric Inequalities
Abstract: TBA If $\{\alpha_k\} \subset \mathbb R^d$ consists of a simplex and a single interior point, and if you impose the condition that $p(x) = \sum c_kx^{\alpha_k}$ vanishes to the second order at $\underbar{1} = (1,\dots,1)$ and is definite, then the resulting polynomial is, up to a multiple, a version of the arithmetic- geometric inequality for the monomials $\{x^{\alpha_k}\}$. There is a sense in which this is an iff condition. In this talk, we explore geometric conditions on larger point-sets $\{\alpha_k\}$ so that imposing a higher even-order vanishing at $\underbar{1}$ leads to an inequality, and present a few preliminary results and a lot of pictures. A riddle, wrapped in a mystery, inside an enigma.