Holomorphic certification on real subspaces

Abstract

We study a family of nonnegative Hermitian quartics on complexifications of real linear subspaces of dimension at least two. A certificate expresses the quartic, after multiplication by a power of the squared Euclidean norm, as a sum of squared absolute values of holomorphic polynomials. We characterize certificate existence by a simple geometric condition: the real subspace contains no vector whose coordinates are all plus or minus one. When such a vector exists, every nonzero intrinsic polynomial multiplier is excluded. This classification yields coNP-hardness even for integer quartics with polynomially bounded coefficients and exactly one negative coefficient eigenvalue. For fixed ambient and proper subspace dimensions and a fixed positive parameter, the least certificate degree grows as the inverse square of the distance to the failure set as that distance tends to zero. We identify that distance exactly in the operator norm on subspace projections. A parallel law holds for bounded matrices imposing soft constraints. The lower bounds use logarithmic curvature; classical quantitative stabilization supplies matching upper bounds. A weighted two-dimensional family gives the exact leading constant and an example with short rational input but exponentially growing certificate degree.