Regular semi-algebraic system
Encyclopedia
In computer algebra, a regular semi-algebraic system is a particular kind of triangular system of multivariate polynomials over a real closed field.
s and triangular decomposition
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
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 
.
be a regular chain
of
for some ordering of the variables
and a real closed field
.
Let
and 
designate respectively the variables of
that are free and algebraic with respect to 
. Let 
be finite such that
each polynomial in
is regular w.r.t.\ the saturated ideal of 
.
Define
. Let 
be a quantifier-free
formula of
involving only the variables of 
. We say that 
is a regular semi-algebraic system if the following three conditions hold.
The zero set of
, denoted by 
,
is defined as the set of points
such that 
is true and 
, 
, for all 
and all 
.
Introduction
Regular chainRegular 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 :...
s and triangular decomposition
Triangular decomposition
In computer algebra, a triangular decomposition of a polynomial system S is a set of simpler polynomial systems S_1,\ldots, S_e such that a point is a solution of S if and only if it is a solution of one of the systems S_1,\ldots, S_e....
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
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 .Formal definition
Let be a regular chainRegular 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 :...
of
for some ordering of the variables
and a real closed fieldReal 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.-Definitions:...
.Let
and designate respectively the variables of
that are free and algebraic with respect to . Let be finite such thateach polynomial in
is regular w.r.t.\ the saturated ideal of .Define
. Let be a quantifier-freeformula of
involving only the variables of . We say that is a regular semi-algebraic system if the following three conditions hold.-
defines a non-empty open semi-algebraic set 
of 
, - the regular system
specializes well at every point 
of 
, - at each point
of 
, the specialized system 
has at least one real zero.
The zero set of
, denoted by ,is defined as the set of points
such that is true and , , for all and all .See also
- Characteristic setCharacteristic setWenjun Wu's method is an algorithm for solving multivariate polynomial equations introduced in the late 1970s by the Chinese mathematician Wen-Tsun Wu. This method is based on the mathematical concept of characteristic set introduced in the late 1940s by J.F. Ritt...
- Triangular decompositionTriangular decompositionIn computer algebra, a triangular decomposition of a polynomial system S is a set of simpler polynomial systems S_1,\ldots, S_e such that a point is a solution of S if and only if it is a solution of one of the systems S_1,\ldots, S_e....
- Regular chainRegular chainIn 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 :...
- RegularChainsRegularChainsThe RegularChains package in the computer algebra software package Maple is a collection of commands for solving systems of polynomial equations, inequations and inequalities symbolically....
- Real algebraic geometryReal algebraic geometryIn mathematics, real algebraic geometry is the study of real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them ....