The obvious inner product for homogeneous polynomials

There is an obvious, and frequently rediscovered, inner product for the vector space of homogeneous polynomials in n variables and fixed degree. The inner product has its roots in 19th century mathematics. Analysts called it the Fisher inner product. Algebraic geometers used it to define apolarity. It arises in many applications ranging from combinatorics and number theory to moment problems and numerical analysis. I will discuss some of the basic properties and present a menu of opportunities for future seminars.