SDP Feasibility Problems and sos Representation Ranks for OT-FKM Type Isoparametric Polynomials

TL;DR

This paper investigates the sum-of-squares representations of specific quartic forms linked to OT-FKM type isoparametric polynomials, characterizing their sos property via semidefinite programming feasibility and providing rank bounds.

Contribution

It characterizes sos properties of certain quartic forms using explicit SDP feasibility conditions and establishes rank bounds for sos representations, especially for m >= 3.

Findings

Sos property characterized by SDP feasibility

Quantitative rank bounds for sos representations

Rigidity results for m >= 3

Abstract

Semidefinite programming (SDP) provides a fundamental framework for studying properties of sum-of-squares (sos) representations of nonnegative polynomials. In this paper we study the quartic forms GF = (|x|^4 + F(x))/2 associated with isoparametric polynomials F of OT-FKM type with g = 4. We characterize the sos property of GF in terms of the feasibility of an explicit SDP determined by the underlying Clifford system, and in the sos cases we obtain quantitative rank bounds for sos representations, with rigidity when m >= 3.