Multiply positive functions, critical exponent phenomena, and the Jain-Karlin-Schoenberg kernel

We prove the converse to a result of Karlin [Trans. AMS 1964], and also strengthen his result and two results of Schoenberg [Ann. of Math. 1955]. One of the latter results concerns zeros of Laplace transforms of multiply positive functions. The other results study which powers $\alpha$ of two specific kernels are totally non-negative of order $p\geq 2$ (denoted TN$_p$); both authors showed this happens for $\alpha\geq p-2$, and Schoenberg proved that it does not for $\alpha<p-2$. We show more strongly that for every $p \times p$ submatrix of either kernel, up to a 'shift', its $\alpha$th power is totally positive of order $p$ (TP$_p$) for every $\alpha > p-2$, and is not TN$_p$ for every non-integer $\alpha\in(0,p-2)$. In particular, these results reveal 'critical exponent' phenomena in total positivity. We also prove the converse to a 1968 result of Karlin, revealing yet another critical exponent phenomenon - for Laplace transforms of all Polya Frequency (PF) functions. We further classify the powers preserving all TN$_p$ Hankel kernels on intervals, and isolate individual kernels encoding these powers. We then transfer results on preservers by Polya-Szego (1925), Loewner/Horn (1969), and Khare-Tao (in press), from positive matrices to Hankel TN$_p$ kernels. Another application constructs individual matrices encoding the Loewner convex powers. This complements Jain's results (2020) for Loewner positivity, which we strengthen to total positivity. Remarkably, these (strengthened) results of Jain, those of Schoenberg and Karlin, the latter's converse, and the above Hankel kernels all arise from a single symmetric rank-two kernel and its powers: $\max(1+xy,0)$. We also provide a novel characterization of PF functions and sequences of order $p\geq 3$, following Schoenberg's 1951 result for $p=2$. We correct a small gap in his paper, in the classification of discontinuous PF functions.