Gårding's Theorem for Posynomials

What the math gives to ML

The paper extends Gårding-type concavity from homogeneous polynomials to homogeneous posynomials with arbitrary nonnegative real exponents. Its key transferable asset is that zero-freeness on a complex sector implies concavity of a fractional power of the posynomial, giving a principled family of diminishing-returns objectives. A practical neural-network use is an MoE router regularizer that rewards balanced expert utilization through a concave geometric-mean utility rather than an entropy penalty. The resulting objective is differentiable, cheap, and can incorporate hardware capacities or expert groups through positive coefficients.