I have a polynomial of degree $6$ with positive real roots. I need to know the conditions on its coefficients so that the polynomial is positive for any positive root, i.e.,
$$a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 > 0$$
and all $x_i > 0$
Ulrich and Watson found the conditions for quartic polynomials (SIAM J. Sci. Comput. 15, 3, 528-544, 1994). But, I don't find them for higher degree polynomials.