  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.
