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# Apolar Schemes of Algebraic Forms

Published:2006-06-01
Printed: Jun 2006
• Jaydeep Chipalkatti
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## Abstract

This is a note on the classical Waring's problem for algebraic forms. Fix integers $(n,d,r,s)$, and let $\Lambda$ be a general $r$-dimensional subspace of degree $d$ homogeneous polynomials in $n+1$ variables. Let $\mathcal{A}$ denote the variety of $s$-sided polar polyhedra of $\Lambda$. We carry out a case-by-case study of the structure of $\mathcal{A}$ for several specific values of $(n,d,r,s)$. In the first batch of examples, $\mathcal{A}$ is shown to be a rational variety. In the second batch, $\mathcal{A}$ is a finite set of which we calculate the cardinality.}
 Keywords: Waring's problem, apolarity, polar polyhedron
 MSC Classifications: 14N05 - Projective techniques [See also 51N35] 14N15 - Classical problems, Schubert calculus

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