I'm not a mathematician, but my research recently led to a mathematical question that seems so clean I feel it must've been studied before. I haven't been able to solve it or find relevant existing work, though, so I thought I'd ask it here.

The question's about functions that are power series with all nonnegative coefficients. To be explicit, I mean functions of the form

$f(x) = \sum_{n=0}^\infty a_n x^n$, where $a_n \ge 0$.

The set of functions of this form is closed under composition - if $f$ and $g$ are of this form, then so is $f \circ g$. My question's about the ways one such function can be factored into two: given an $f$ of this form, what $g$, $h$ also of this form can we find such that $g \circ h = f$?

It's easy to see there's always a trivial solution where either $f$ or $g$ has only a linear term; we can always take $g(x) = a_1 x$ and $h(x) = f(x)/a_1$, for example. Sometimes, though, there's no nontrivial solution! For example, there's no way to nontrivially factor $f(x) = x + x^2$ into two other "positive-semidefinite" polynomials.

Is there a simple way to characterize the set of ways to factor a given "positive-semidefinite" polynomial, or even just a simple way to determine if a given PSD polynomial can be split nontrivially? A bonus question I'm also curious about is whether there's an easy way to find the "square-root" of a given PSD polynomial (i.e. given $f$, find a $g$ where $g \circ g = f$).



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