In computer algebra, a regular semi-algebraic system is a particular kind of triangular system of multivariate polynomials over a real closed field.
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
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
{\displaystyle \mathbf {y} =y_{1},\ldots ,y_{n-d}}
designate respectively the variables of
that are free and algebraic with respect to
be finite such that each polynomial in
is regular with respect to the saturated ideal of
Define
be a quantifier-free formula of
involving only the variables of
is a regular semi-algebraic system if the following three conditions hold.
The zero set of
, is defined as the set of points
{\displaystyle (u,y)\in \mathbf {k} ^{d}\times \mathbf {k} ^{n-d}}
is true and
Observe that
has dimension
in the affine space
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