Positivity and nonnegativity conditions of a quartic equation and related problems

In this paper explicit conditions for positivity (no real roots), nonnegativity on positive real axis (no positive real roots with odd multiplicity), and stability aperiodicity (all roots are real, and, negative and simple) of a quartic (or biquadratic) equation are given. The derived conditions from the known solution of the quartic equation are not only complete, but simpler than those derived from Sturm, extended Hurwitz, inners, and Hankel methods. Because of Abel´s Theorem (no explicit solution in terms of the roots of an equation higher than quartic exists), similar simplification for higher degree polynomial equations may not be possible. Furthermore, explicit conditions for positivity and nonnegativity of equations of higher degree than four are extremely difficult to obtain and may not be possible. The results of the paper will hopefully shed some light on a century old problem and thus enhance the engineering application of the derived condition to higher order systems.