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.
Regular chains 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 formula_1 can be decomposed into finitely many regular semi-algebraic systems formula_2 such that a point (with real coordinates) is a solution of formula_1 if and only if it is a solution of one of the systems formula_2.
Formal definition.
Let formula_5 be a regular chain of formula_6 for some ordering of the variables formula_7 and a real closed field formula_8. Let formula_9 and formula_10 designate respectively the variables of formula_11 that are free and algebraic with respect to formula_5. Let formula_13 be finite such that each polynomial in formula_14 is regular with respect to the saturated ideal of formula_5. Define formula_16. Let formula_17 be a quantifier-free formula of formula_18 involving only the variables of formula_19. We say that formula_20 is a regular semi-algebraic system if the following three conditions hold.
The zero set of formula_30, denoted by formula_31, is defined as the set of points formula_32 such that formula_33 is true and formula_34, for all formula_35and all formula_36. Observe that formula_31 has dimension formula_38 in the affine space formula_39.