Consider a univariate polynomial of degree $n$ with real coefficients. Are there general equalities/inequalities on its coefficients, so that it has precisely $n$ real roots?
For example for the case of $n=2,3$ the necessary and sufficient inequality is given by the discriminant of the polynomial. If the discriminant is non-negative, the polynomial has only real roots. The case $n=4$ can also be handled since formulas for the roots exist. How about for larger $n$?