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Abstract

For normalized Lorentzian polynomials of degree n in n variables, we prove that the power of a linear form with prescribed gradient is the unique minimizer of the squarefree coefficient whenever that gradient is sufficiently close to the all-ones vector. The coefficient excess controls the full coefficient ℓ¹ distance, uniformly in each fixed degree. The proof gives an explicit separation bound for the cone determined by the transverse quadratic derivatives at the balanced power. This bound also yields an explicit local stability estimate when both gradient and polynomial are close to that power. Variable splitting extends the result to positive integer compositions, and volume polynomials give a mixed-volume coefficient stability bound.