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In [[computer algebra]], a '''regular semi-algebraic system''' is a particular kind of triangular  system  of multivariate polynomials over a real closed field.
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== Introduction ==
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[[Regular chain]]s and [[triangular decomposition]]s 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 <math>S</math> can be decomposed into finitely many regular semi-algebraic systems <math>S_1, \ldots, S_e</math> such that a point (with real coordinates) is a solution of <math>S</math> if and only if it is a solution of one of the systems <math>S_1, \ldots, S_e</math>.<ref>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.</ref>
 
== Formal definition ==
 
Let <math>T</math> be a [[regular chain]] of <math>{{\mathbf{k}}}[x_1, \ldots, x_n]</math>
for some ordering of the variables <math>\mathbf{x} = x_1, \ldots, x_n</math> and a [[real closed field]] <math>{{\mathbf{k}}}</math>.
Let <math>\mathbf{u} = u_1, \ldots, u_d</math> and <math>\mathbf{y} = y_1, \ldots, y_{n-d}</math>
designate respectively the variables of <math>\mathbf{x}</math> that are free and algebraic with respect to <math>T</math>. Let <math>P \subset {{\mathbf{k}}}[\mathbf{x}]</math> be finite such that
each polynomial in <math>P</math> is regular w.r.t.\ the saturated ideal of <math>T</math>.
Define <math>P_{>} :=\{p>0\mid p\in P\}</math>. Let <math>\mathcal{Q}</math> be a quantifier-free
formula of <math>{{\mathbf{k}}}[\mathbf{x}]</math> involving only the variables of <math>\mathbf{u}</math>. We say that <math>R := [\mathcal{Q}, T, P_{>}]</math> is a '''regular semi-algebraic system''' if the following three conditions hold.
 
* <math>\mathcal{Q}</math> defines a non-empty open semi-algebraic set <math>S</math> of <math>{{\mathbf{k}}}^d</math>,
* the regular system <math>[T, P] </math> specializes well at every point  <math>u</math> of <math>S</math>,
* at each point <math>u</math> of <math>S</math>, the specialized system <math>[T(u), P(u)_{>}</math> has at least one real zero.
 
The zero set of <math>R</math>, denoted by <math>{Z}_{{{\mathbf{k}}}}(R)</math>,
is defined as the set of points <math>(u, y) \in {{{\mathbf{k}}}}^d \times {{{\mathbf{k}}}}^{n-d}</math> such that <math>\mathcal{Q}(u)</math> is true and  <math>t(u, y)=0</math>,  <math>p(u, y)>0</math>,  for all <math>t\in T</math>and all <math>p\in P</math>.
 
== See also ==
*[[Wu's method of characteristic set|Characteristic set]]
*[[Triangular decomposition]]
*[[Regular chain]]
*[[RegularChains]]
*[[Real algebraic geometry]]
 
== References ==
{{Reflist}}
 
[[Category:Equations]]
[[Category:Algebra]]
[[Category:Polynomials]]
[[Category:Algebraic geometry]]
[[Category:Computer algebra]]

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