Lower Bounds for Positive-Power Exponents in a Hermitian Polynomial Family

For a fixed integer \(k\ge2\), let \(N_k(t)\) be the least positive integer \(n\) for which \(((1+x)^{2k}-tx^k)^n\) has nonnegative coefficients. Known positivity theorems imply that \(N_k(t)\) is finite for \(0\)\({}\le t\)\({}\lt T_k\)\({}=2^{2k-1}\), whereas no exponent works at \(t=T_k\). We give an explicit necessary lower bound as \(t\) approaches this boundary. Writing \(\varepsilon=T_k-t\), \(L\)\({}=\log(1\)\(/\varepsilon)\), and \(\beta_k\)\({}=T_k(k\)\({}-1)\)\(/(4k)\), we prove \[\liminf_{\varepsilon\downarrow0}\bigl[\varepsilon N_k(T_k-\varepsilon)-\beta_k L-\beta_k\log L\bigr]\ge C_k,\] with an explicit Gamma-function constant \(C_k\).

The proof compares a Gaussian contribution to a selected coefficient with a negative contribution flat to order \(2k\). The estimates are uniform in \(t\) near the boundary. Equivalently, they bound the exponent needed to express a power of a known bihomogeneous Hermitian form as a holomorphic squared norm. No matching upper bound or general criterion for powers of Hermitian polynomials is asserted. This AI-assisted preprint is self-audited and unrefereed; it does not resolve the general AIM problem, and absolute priority is not certified.