Regular Semi-algebraic System
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In
computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expression ...
, a regular semi-algebraic system is a particular kind of triangular system of multivariate polynomials over a real closed field.


Introduction

Regular chain In computer algebra, a regular chain is a particular kind of triangular set in a multivariate polynomial ring over a field. It enhances the notion of characteristic set. Introduction Given a linear system, one can convert it to a triangular ...
s 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 In computer algebra, a regular chain is a particular kind of triangular set in a multivariate polynomial ring over a field. It enhances the notion of characteristic set. Introduction Given a linear system, one can convert it to a triangular ...
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 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/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 Tand 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