Semi-algebraic Complexity of Quotients and Sign Determination of Remainders

For two nonzero polynomialsformula]the successive signed Euclidean division yields algorithms, that is, semialgebraic computation trees, for tasks such as computing the sequence of successive quotients, deciding the signs of the sequence of remainders in a pointa∈R, deciding the number of remainders, or deciding the degree pattern of the sequence of remainders. In this paper we show lower bounds of ordermlog₂(m) for these tasks, within the computational framework of semi-algebraic computation trees. The inevitably long paths in semi-algebraic computation trees can be specified as those followed by certain prime cones in the real spectrum of a polynomial ring. We give in the paper a positive answer to the question posed in T. Lickteig (J. Pure Appl. Algebra110(2), 131–184 (1996)) whether the degree of the zero-set of the support of a prime cone provides a lower bound on the complexity of isolating the prime cone. The applications are based on the degree inequalities given by Strassen and Schuster and extend their work to the real situation in various directions.