Given a domain $I\subset\Bbb{C}$ and an integer $N>0$, a function $f: I\to\Bbb{C}$ is said to be {\it entrywise positivity preserving} on positive semidefinite $N\times N$ matrices $A=(a_{jk})\in I^{N\times N}$, if the entrywise application $f[A]=(f(a_{jk}))$ of $f$ to $A$ is positive semidefinite for all such $A$. Such preservers in all dimensions have been classified by Schoenberg and Rudin as being absolutely monotonic. In fixed dimension $N$, results akin to work of Horn and Loewner show that the first $N$ non-zero Maclaurin coefficients of any positivity preserver $f$ are positive; and the last $N$ coefficients are also positive if $I$ is unbounded. However, very little was known about the higher-order coefficients: the only examples to date for unbounded domains $I$ were absolutely monotonic, hence work in all dimensions; and for bounded $I$ examples of non-absolutely monotonic preservers were very few (and recent).

In this paper, we provide a complete characterization of the sign patterns of the higher-order Maclaurin coefficients of positivity preservers in fixed dimension $N$, over bounded and unbounded domains $I=(0,\rho)$. In particular, this shows that the above Horn--Loewner-type conditions cannot be improved upon. As a further special case, this provides the first examples of polynomials which preserve positivity on positive semidefinite matrices in $I^{N\times N}$ but not in $I^{(N+1)\times (N+1)}$. Our main tools in this regard are the Cauchy--Binet formula and lower and upper bounds on Schur polynomials. We also obtain analogous results for real exponents, using the Harish-Chandra--Itzykson--Zuber formula in place of bounds on Schur polynomials.

We then go from qualitative existence bounds---which suffice to understand all possible sign patterns---to exact quantitative bounds. This is achieved using a Schur positivity result due to Lam, Postnikov, and Pylyavskyy, and in particular provides a second proof of the existence of threshold bounds for tuples of integer and real powers. 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 of Cuttler, Greene, and Skandera to obtain a novel characterization of weak majorization for real tuples.


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pp. 1863-1929
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