Regular semi-algebraic system explained

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 can be decomposed into finitely many regular semi-algebraic systems S _{1,} \ldots, 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,} \ldots, S _{e} .^[1]

Formal definition

Let

T be a regular chain of k [x _{1,} \ldots, x _{n]} for some ordering of the variables x = x _{1,} \ldots, x _{n} and a real closed field k . Let u = u _{1,} \ldots, u _{d} and y = y _{1,} \ldots, y _{n-d} designate respectively the variables of x that are free and algebraic with respect to T . Let P \subset k [ x ] be finite such that each polynomial in P is regular with respect to the saturated ideal of T . Define P _{>} : = \{p> 0 \mid p\in P\} . Let l{Q} be a quantifier-free formula of k [ x ] involving only the variables of u . We say that R : = [ l{Q}, T, P _{>} ] is a regular semi-algebraic system if the following three conditions hold.l{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) \in k ^{d} x k ^{n-d} such that l{Q}(u) is true and t(u, y) =0 , p(u, y)> 0 , for all t\in T and all p\in P . Observe that Z _{k} (R) has dimension d in the affine space k ^{n} .

See also

• Real algebraic geometry

Notes and References

1. 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.