Regular semi-algebraic system

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 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 .[1]

Formal definition

Let 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 with respect to 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.

  • 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 . Observe that has dimension in the affine space .

See also

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.


This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.