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 $S_{1},\ldots ,S_{e}$.^[1]

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 w.r.t.\ 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 ${\displaystyle {\mathbf {k} }^{n}}$.

See also

• RegularChains
• Real algebraic geometry

References

1. Changbo Chen, James H. Davenport, John P. May, Marc Moreno-Maza, Bican Xia, Rong Xiao. Triangular decomposition of semi-algebraic systems. Proceedings of 2010 International Symposium on Symbolic and Algebraic Computation (ISSAC 2010), ACM Press, pp. 187–194, 2010.