Apolar Schemes of Algebraic Forms

Source abstract

Abstract This is a note on the classical Waring's problem for algebraic forms. Fix integers ( n , d , r , s ), and let ∧ be a general r -dimensional subspace of degree d homogeneous polynomials in n +1 variables. Let denote the variety of s -sided polar polyhedra of ∧. We carry out a case-by-case study of the structure of for several specific values of ( n , d , r , s ). In the first batch of examples, is shown to be a rational variety. In the second batch, is a finite set of which we calculate the cardinality.