An improvement of the projection operator in cylindrical algebraic decomposition

The Cylindrical Algebraic Decomposition (CAD) method of Collins [5] decomposes <italic>r</italic>-dimensional Euclidean space into regions over which a given set of polynomials have constant signs. An important component of the CAD method is the projection operation: given a set <italic>A</italic> of <italic>r</italic>-variate polynomials, the projection operation produces a set <italic>P</italic> of (<italic>r</italic> - 1)-variate polynomials such that a CAD of <italic>r</italic>-dimensional space for <italic>A</italic> can be constructed from a CAD of (<italic>r</italic>-1)-dimensional space for <italic>P</italic>. In this paper, we present an improvement to the projection operation. By generalizing a lemma on which the proof of the original projection operation is based, we are able to find another projection operation which produces a smaller number of polynomials. Let <italic>m</italic> be the number of polynomials contained in <italic>A</italic>, and let <italic>n</italic> be a bound for the degree of each polynomial in <italic>A</italic> in the projection variable. The number of polynomials produced by the original projection operation is dominated by <italic>m</italic><supscrpt>2</supscrpt><italic>n</italic><supscrpt>3</supscrpt> whereas the number of polynomials produced by our projection operation is dominated by <italic>m</italic><supscrpt>2</supscrpt><italic>n</italic><supscrpt>2</supscrpt>.