I don't need to know solutions. I figured that the logic would be similar to computing the discriminant and testing whether it is positive. For example in quadratic systems with real coefficients,
$$ ax^2 + bx + c = 0$$
if $b^2 - 4ac < 0$ then both solutions are complex, otherwise they are real. Then, if c > 0 they are the same sign, and then the sign of $b$ determines what their sign is if so.
Thus if $b^2 - 4ac > 0$ and $c < 0$ I know the quadratic equation has exactly one real, positive solution.
Are there general versions of this criteria?
You should look at Sturm's Theorem which tells you the number of roots a polynomial has in an interval. Taking a lower limit $0$ and an upper limit sufficiently large to be beyond any real zero you can check the number of positive real solutions.
This is probably not quite the kind of answer you were looking for, but it does work for polynomials of arbitrary degree.