Regular semi-algebraic system


In computer algebra, a regular semi-algebraic system is a particular kind of triangular system of multivariate polynomials over a real closed field.

Introduction

s and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. The notion of a regular semi-algebraic system is an adaptation of the concept of a regular chain focusing on solutions of the real analogue: semi-algebraic systems.
Any semi-algebraic system can be decomposed into finitely many regular semi-algebraic systems such that a point is a solution of if and only if it is a solution of one of the systems.

Formal definition

Let be a regular chain of for some ordering of the variables and a real closed field. Let and designate respectively the variables of that are free and algebraic with respect to. Let be finite such that each polynomial in is regular with respect to the saturated ideal of. Define. Let be a quantifier-free formula of involving only the variables of. We say that is a regular semi-algebraic system if the following three conditions hold.
The zero set of, denoted by, is defined as the set of points such that is true and, for all and all. Observe that has dimension in the affine space.