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 $$S$$ can be decomposed into finitely many regular semi-algebraic systems $$S_1, \ldots, S_e$$ such that a point (with real coordinates) is a solution of $$S$$ if and only if it is a solution of one of the systems $$S_1, \ldots, S_e$$.

Formal definition
Let $$T$$ be a regular chain of $$\mathbf{k}[x_1, \ldots, x_n]$$ for some ordering of the variables $$\mathbf{x} = x_1, \ldots, x_n$$ and a real closed field $$\mathbf{k}$$. Let $$\mathbf{u} = u_1, \ldots, u_d$$ and $$\mathbf{y} = y_1, \ldots, y_{n-d}$$ designate respectively the variables of $$\mathbf{x}$$ that are free and algebraic with respect to $$T$$. Let $$P \subset \mathbf{k}[\mathbf{x}]$$ be finite such that each polynomial in $$P$$ is regular with respect to the saturated ideal of $$T$$. Define $$P_{>} :=\{p>0\mid p\in P\}$$. Let $$\mathcal{Q}$$ be a quantifier-free formula of $$\mathbf{k}[\mathbf{x}]$$ involving only the variables of $$\mathbf{u}$$. We say that $$R := [\mathcal{Q}, T, P_{>}]$$ is a regular semi-algebraic system if the following three conditions hold.


 * $$\mathcal{Q}$$ defines a non-empty open semi-algebraic set $$S$$ of $$\mathbf{k}^d$$,
 * the regular system $$[T, P]$$ specializes well at every point $$u$$ of $$S$$,
 * at each point $$u$$ of $$S$$, the specialized system $$[T(u), P(u)_{>}]$$ has at least one real zero.

The zero set of $$R$$, denoted by $$Z_{\mathbf{k}}(R)$$, is defined as the set of points $$(u, y) \in \mathbf{k}^d \times \mathbf{k}^{n-d}$$ such that $$\mathcal{Q}(u)$$ is true and $$t(u, y)=0, p(u, y)>0$$, for all $$t\in T$$and all $$p\in P$$. Observe that $$Z_{\mathbf{k}}(R)$$ has dimension $$d$$ in the affine space $$\mathbf{k}^n$$.