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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 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 ...
s and
triangular decomposition In computer algebra, a triangular decomposition of a polynomial system is a set of simpler polynomial systems such that a point is a solution of if and only if it is a solution of one of the systems . When the purpose is to describe the solut ...
s 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 s ...
of \mathbf _1, \ldots, x_n/math> for some ordering of the variables \mathbf = x_1, \ldots, x_n and a
real closed field In mathematics, a real closed field is a field ''F'' that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. D ...
\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 Tand all p\in P. Observe that Z_(R) has dimension d in the affine space \mathbf^n.


See also

*
Real algebraic geometry In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial ...


References

{{Reflist Equations Algebra Polynomials Algebraic geometry Computer algebra