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Sums of squares of k-term forms

Charu Goel, Bruce Reznick

Abstract

In this paper we study the cones corresponding to sums of squares of $n$-ary $d$-ic forms with at most $k$ terms. We show that these are strictly nested as $k$ increases, leading to the usual sum of squares cone. We also discuss the duals of these cones. For $n \ge 3$, we construct indefinite irreducible $n$-ary $d$-ic forms with exactly $k$ terms for $2 \le k \le \binom{n+d-1}{n-1}$.

Sums of squares of k-term forms

Abstract

In this paper we study the cones corresponding to sums of squares of -ary -ic forms with at most terms. We show that these are strictly nested as increases, leading to the usual sum of squares cone. We also discuss the duals of these cones. For , we construct indefinite irreducible -ary -ic forms with exactly terms for .
Paper Structure (3 sections, 11 theorems, 58 equations)

This paper contains 3 sections, 11 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. The cone its dual, and the proof of Theorem \ref{['T:thmA']}
  3. Proof of Theorem \ref{['T:thmB']}

Key Result

Theorem 1.1

For all $n,d,k$, $\Sigma_{n,2d}^{k}$ is a closed convex cone.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Corollary 1.4
  • proof
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['T:thmA']}
  • Theorem 2.2
  • Lemma 2.3
  • ...and 9 more