Quantifier Elimination for Quartics
Concerning quartics, two particular quantifier elimination (QE) problems of historical interests and practical values are studied. We solve the problems by the theory of complete discrimination systems and negative root discriminant sequences for polynomials that provide a method for real (positive/negative) and complex root classification for polynomials. The equivalent quantifier-free formulas are obtained mainly be hand and are simpler than those obtained automatically by previous methods or QE tools. Also, applications of the results to program verification and determination of positivity of symmetric polynomials are showed.