Reformulating the SOS conjecture via Newton-Okounkov bodies shows the minimal rank occurs at extreme points of the body, and for diagonal polynomials these reduce to finitely many rational points.
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A Newton-Okounkov Body Viewpoint on the SOS Conjecture
Reformulating the SOS conjecture via Newton-Okounkov bodies shows the minimal rank occurs at extreme points of the body, and for diagonal polynomials these reduce to finitely many rational points.