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
can be decomposed into finitely many regular semi-algebraic systems
such that a point (with real coordinates) is a solution of
if and only if it is a solution of one of the systems
.
[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
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