A Finite-order Characterization of Entrywise Positivity Preservers
Fix $I = (0,\rho)$, where $0<\rho\leq\infty$, and let $\mathbb{P}_n(I)$ be the set of positive semidefinite $n\times n$ matrices with entries in $I$. A longstanding problem in matrix theory is to characterize the functions $f: I \to \mathbb{R}$ for which the entrywise calculus $f[A] = [f(a_{ij})]_{i,j = 1}^{n}$ preserves positive semidefiniteness for all $A \in \mathbb{P}_n(I)$. We characterize these functions exactly: if $f\in C^{2n-2}(I)$ and $\mathcal{E} = x\frac{d}{dx}$, then this holds if and only if $$ f^{(k)}(x)\geq 0 \quad (0\leq k\leq n-1) \qquad\text{and}\qquad \bigl[\mathcal{E}^{i+j}f(x)\bigr]_{i,j = 0}^{n-1}\succeq 0 $$ for every $x\in I$. Regularization then removes all a priori smoothness: for $n\geq2$, every preserver belongs to $C^{2n-4}(I)$ and the same characterization holds by interpreting the last two derivatives in the sense of distributions. As applications, we recover classical results of FitzGerald--Horn and Vasudeva, and obtain a complete classification of generalized polynomials with prescribed real exponents and arbitrary coefficients. We also determine optimal constants in entrywise domination inequalities under finite regularity, extend the sharp finite-sum thresholds of Belton--Guillot--Khare--Putinar and Khare--Tao to positive mixtures of powers, and answer a question of Khare and Tao by showing that no finite collection of matrices with entries strictly inside $I$ can detect positivity preservation on $\mathbb{P}_n(I)$.