2025/12/08 by Wang, Zhiwei, Yue, Chenlong, Zhou, Xiangyu
#Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2512.07133
Let z∈ \mathbb Cn be the complex coordinates on \mathbb Cn, and A(z, z) be a real-valued Hermitian polynomial. The famous Ebenfelt's SOS conjecture asks for the minimum rank of A(z, z)‖z‖2 under the restriction that A(z, z)‖z‖2 is an SOS. Assume that A(z, z) is bihomogeneous. In the present note, we establish a connection between Ebenfelt's (Weak) SOS Conjecture and the theory of Newton-Okounkov bodies. By reformulating the conjecture in terms of lattice semigroups and their associated Newton-Okounkov convex bodies, we transform the problem of finding the minimal rank of a prolonged sum-of-squares polynomial into an extremal problem in convex geometry. In particular, we prove that this minimal rank is attained at the extreme points of a specific Newton-Okounkov body. Furthermore, if A(z, z) is moreover diagonal, we demonstrate that the relevant extreme points are finitely many rational points, thereby reducing the verification of the conjecture to a computationally tractable problem. This work provides a new tool for attacking the SOS Conjecture.