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