Holomorphic certification on real subspaces

Abstract

We study when a nonnegative Hermitian quartic enters the hierarchy of holomorphic squared-norm certificates obtained by multiplication by powers of its squared norm. Deciding eventual membership is coNP-hard even for integer quartics with polynomially bounded coefficients and exactly one negative coefficient eigenvalue. A sign-cube criterion on complexifications of real subspaces supplies the reduction: avoiding the sign cube gives stabilization, while a sign vector excludes every intrinsic nonzero polynomial multiplier. We also quantify the degree cost near this geometric boundary. For bounded real matrices imposing soft linear constraints, the least degree grows uniformly as the inverse square of the smallest signed constraint residual. A logarithmic curvature obstruction gives the lower bound, matched by quantitative classical stabilization. On a weighted two-dimensional hyperplane family, complete finite-matrix bounds determine the exact leading constant. These results connect a restricted computational decision with both a uniform boundary scale and a sharp fixed-dimensional example.