Abstract:
The weak real Nullstellensatz gives an algebraic certificate for a geometric property of a set of polynomials, namely, having no common zeroes over the real numbers. The effective real Nullstellensatz seeks for an efficient way of computing such a certificate. One of the main methods to obtain these certificates is what we know as quantifier elimination. Cylindrical algebraic decomposition is an algorithm designed to perform quantifier elimination over the real polynomials. In particular, the algorithm can decide whether a finite set of polynomials has common zeroes over the real numbers or not. The main question in this work, is the applicability of the cylindrical algebraic decomposition algorithm for achieving the effective weak real Nullstellensatz. In this work, we first develop the theory behind the weak real Nullstellensatz and discuss the recent developments in obtaining effective versions of it. Second, we describe cylindrical algebraic decomposition. Most importantly, we examine a method to use cylindrical algebraic decomposition for the effective weak real Nullstellensatz. We do this by introducing simpler weak real Nullstellensatz certificates which lead us to the desired certificate. Finally, we discuss some of the difficulties that one faces in using cylindrical algebraic decomposition for obtaining the effective weak real Nullstellensatz.