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 $S_1, \ldots, S_e$.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.

Formal definition

Let $T$ be a regular chain of \mathbf _1, \ldots, x_n/math> for some ordering of the variables $\mathbf = x_1, \ldots, x_n$ and a real closed field $\mathbf$. Let $\mathbf = u_1, \ldots, u_d$ and $\mathbf = y_1, \ldots, y_$ designate respectively the variables of $\mathbf$ that are free and algebraic with respect to $T$. Let P \subset \mathbf mathbf/math> be finite such that each polynomial in $P$ is regular with respect to the saturated ideal of $T$. Define $P_ :=\$. Let $\mathcal$ be a quantifier-free formula of \mathbf mathbf/math> involving only the variables of $\mathbf$. We say that R := mathcal, T, P_/math> is a regular semi-algebraic system if the following three conditions hold.

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

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

See also

*Real algebraic geometry

References

{{Reflist

Equations
Algebra
Polynomials
Algebraic geometry
Computer algebra