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

Formal definition

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

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

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

See also

• Real algebraic geometry