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 S can be decomposed into finitely many regular semi-algebraic systems S 1 , … , 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 , … , S e .
Let T be a regular chain of k [ x 1 , … , x n ] for some ordering of the variables x = x 1 , … , x n and a real closed field k . Let u = u 1 , … , u d and y = y 1 , … , y n − d designate respectively the variables of x that are free and algebraic with respect to T . Let P ⊂ k [ x ] be finite such that each polynomial in P is regular w.r.t.\ the saturated ideal of T . Define P > := { p > 0 ∣ p ∈ P } . Let Q be a quantifier-free formula of k [ x ] involving only the variables of u . We say that R := [ Q , T , P > ] is a regular semi-algebraic system if the following three conditions hold.
Q defines a non-empty open semi-algebraic set S of k d ,the regular system [ T , P ] specializes well at every point u of S ,at each point u of S , the specialized system [ T ( u ) , P ( u ) > ] has at least one real zero.The zero set of R , denoted by Z k ( R ) , is defined as the set of points ( u , y ) ∈ k d × k n − d such that Q ( u ) is true and t ( u , y ) = 0 , p ( u , y ) > 0 , for all t ∈ T and all p ∈ P . Observe that Z k ( R ) has dimension d in the affine space k n .
