Real quantifier elimination is doubly exponential
Journal of Symbolic Computation
Geometric reasoning with logic and algebra
Artificial Intelligence - Special issue on geometric reasoning
Quantifier elimination for formulas constrained by quadratic equations
ISSAC '93 Proceedings of the 1993 international symposium on Symbolic and algebraic computation
Recent advances on determining the number of real roots of parametric polynomials
Journal of Symbolic Computation - Special issue on polynomial elimination—algorithms and applications
Subresultants and Reduced Polynomial Remainder Sequences
Journal of the ACM (JACM)
New structure theorem for subresultants
Journal of Symbolic Computation - Special issue on symbolic computation in algebra, analysis and geometry
Sylvester—Habicht sequences and fast Cauchy index computation
Journal of Symbolic Computation
QEPCAD B: a program for computing with semi-algebraic sets using CADs
ACM SIGSAM Bulletin
Computing real zeros of polynomials with parametric coefficients
ACM SIGSAM Bulletin
An algorithm for computing the complete root classification of a parametric polynomial
AISC'06 Proceedings of the 8th international conference on Artificial Intelligence and Symbolic Computation
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An improved algorithm, together with its implementation, is presented for the automatic computation of the complete root classification of a real parametric polynomial. The algorithm offers improved efficiency and a new test for non-realizable conditions. The improvement lies in the direct use of 'sign lists', obtained from the discriminant sequence, rather than 'revised sign lists'. It is shown that the discriminant sequences, upon which the sign lists are based, are closely related both to Sturm-Habicht sequences and to subresultant sequences. Thus calculations based on any of these quantities are essentially equivalent. One particular application of complete root classifications is the determination of the conditions for the positive definiteness of a polynomial, and here the new algorithm is applied to a class of sparse polynomials. It is seen that the number of conditions for positive definiteness remains surprisingly small in these cases.