Does there exist a polynomial-time algorithm to determine whether a given polynomial $p(n)$ with integer coefficients is positive on $\mathbb{N}$, in the sense that $p(n) \geq 0$ for all $n\in\mathbb{N}$?

This question seems to be related, but it involves multivariable polynomials and asks about positivity on all of $\mathbb{R}^n$, and the answer doesn't seem to apply to single-variable polynomials.

**Edit:** Just to clarify, the input for the algorithm should be a string that encodes the polynomial $p$ in a reasonably compact fashion, e.g. the digits of the degree of $p$ followed by the digits of the coefficients.