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 $\mathbf {k} ^{n}$.

See also

• Real algebraic geometry