On the sign patterns of entrywise positivity preservers in fixed dimension

Given $I\subset\mathbb{C}$ and an integer $N>0$, a function $f:I\to\mathbb{C}$ is entrywise positivity preserving on positive semidefinite (p.s.d.) matrices $A=(a_{jk})\in I^{N\times N}$, if the entrywise application $f[A]=(f(a_{jk}))$ of $f$ to $A$ is p.s.d. for all such $A$. Such preservers in all dimensions have been classified by Schoenberg and Rudin as being absolutely monotonic [Duke Math. J. 1942, 1959]. In fixed dimension $N$, results akin to work of Horn and Loewner [Trans. AMS 1969] show the first $N$ nonzero Maclaurin coefficients of a positivity preserver $f$ are positive; and the last $N$ coefficients are also positive if $I$ is unbounded. However, little was known about the other coefficients: the only examples to date for unbounded domains $I$ were absolutely monotonic, so work in all dimensions; and for bounded $I$ examples of non-absolutely monotonic preservers were very few (and recent). In this paper, we completely characterize the sign patterns of the Maclaurin coefficients of positivity preservers in fixed dimension $N$, over bounded and unbounded domains $I$. In particular, the above Horn-type conditions cannot be improved upon. This also yields the first polynomials which preserve positivity on p.s.d. matrices in $I^{N\times N}$ but not in $I^{(N+1)\times (N+1)}$. We obtain analogous results for real exponents using the Harish-Chandra-Itzykson-Zuber formula. We then go from qualitative bounds, which suffice to understand all possible sign patterns, to exact quantitative bounds. As an application, we extend our previous qualitative and quantitative results to understand preservers of total non-negativity in fixed dimension - including their sign patterns. We deduce several further applications, including extending a Schur polynomial conjecture by Cuttler-Greene-Skandera to obtain a novel characterization of weak majorization for real tuples.