Both algebraically or with software (numerically/computationally, for example) is acceptable. Here is a how one of the polynomials looks like ($L<1$ is a constant).
I have attached a text file with all the 13 polynomials that I need to sort in ascending order - more precisely, I have a target order I need to achieve, so I need to define the ranges of the variables $h_0$ - $h_4$ where that order would be valid. Here is one representative polynomial sample:
$$ 7 h_0 h_1^2 h_2^2 h_3 L^3 + 29 h_0 h_1^2 h_2^2 h_3 h_4^2 L^4 + 3 h_0 h_2^2 h_4 L^2 + 5 h_0 h_2^2 h_4^3 L^3 + \ldots $$
Each polynomial has approximately 200 terms(!).