# Conditions for a power of a polynomial to have no negative coefficients

Consider a polynomial in one variable $$p(x)$$ with $$p(0)>0$$, and that is not a polynomial in $$x^m$$ for any $$m>1$$ (that is, the $$gcd$$ of the exponents appearing in $$p(x)$$ is 1). I would like to find necessary and sufficient conditions so that a power of $$p$$ has no negative coefficients. I know of a sufficient condition: if the degree is $$d$$, then some power of $$p$$ will have no negative coefficients if the coefficients of $$x, x^d$$ and $$x^{d-1}$$ are all positive, AND the inequality $$|p(z)| holds for all complex numbers $$z$$ other than positive real numbers. This condition is obviously not necessary because $$p(x)=1+x^2+x^3$$ does not satisfy it, yet its first power has no negative coefficients. On the other hand the condition $$|p(z)| for non-real and positive $$z$$ by itself is surely necessary (because it is satisfied by polynomials without negative coefficients, so if $$p^n$$ satisfies it, $$p$$ itself must satisfy it, by taking the $$n$$-th root), but not sufficient, as simple examples will show. Other necessary conditions can be obtained by identifying some crucial coefficients towards the beginning and the end of $$p(x)$$ that must be positive (see my article On the inequality $$|p(z)|\leq p(|z|)$$ for polynomials). But there are examples that show that even if all these crucial coefficients are positive and the $$|p(z)| condition holds, the polynomial may still have some negative coefficients in all of its powers. So except for the rather special case when the coefficients of $$x,x^d, x^{d-1}$$ are positive, I do not know how to predict when a power of a polynomial will have no negative coefficients. Obviously my list of "crucial coefficients" that must be positive is not enough but I do not know how to enlarge it.

As a related note, David Handelman has proved that in the (much more difficult) context of polynomials in several variables, if one power of $$p$$ has no negative coefficients, then $$p^n$$ will have no negative coefficients for all sufficiently large $$n$$.