2016/07/01 by Scott McCallum, McCallum, Scott, Adam Parusiński +5 · 1 citation
Computer Science · Engineering · Mathematics · #14P10 #68W30 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #FOS: Mathematics #I.1.2 #Polynomial and algebraic computation #Robotic Mechanisms and Dynamics #acm:14P10 #acm:68W30 #math.AG #msc:14P10 #msc:68W30
paper · pdf · doi:10.48550/arxiv.1607.00264
21 pages
openalex publication_date 2016/07/01 · arxiv created 2017/07/26 · arxiv updated 2017/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 1994 Lazard proposed an improved method for cylindrical algebraic decomposition (CAD). The method comprised a simplified projection operation together with a generalized cell lifting (that is, stack construction) technique. For the proof of the method's validity Lazard introduced a new notion of valuation of a multivariate polynomial at a point. However a gap in one of the key supporting results for his proof was subsequently noticed. In the present paper we provide a complete validity proof of Lazard's method. Our proof is based on the classical parametrized version of Puiseux's theorem and basic properties of Lazard's valuation. This result is significant because Lazard's method can be applied to any finite family of polynomials, without any assumption on the system of coordinates. It therefore has wider applicability and may be more efficient than other projection and lifting schemes for CAD.