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

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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

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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

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References

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  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.