Regular semi-algebraic system

Template:One source 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 $S_1,\dots ,S_e$ 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 $S_1,\dots ,S_e$.

Formal definition

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

• $𝒬$ defines a non-empty open semi-algebraic set $S$ of ${𝐤}^d$,
• 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 $Z_{𝐤}(R)$, is defined as the set of points $(u,y)\in {𝐤}^d\times {𝐤}^{n-d}$ such that $𝒬(u)$ is true and $t(u,y)=0,p(u,y)>0$, for all $t\in T$and all $p\in P$. Observe that $Z_{𝐤}(R)$ has dimension $d$ in the affine space ${𝐤}^n$.

See also

• Real algebraic geometry

References

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