Toric code

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 $S_1,\dots ,S_e$ 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 $S_1,\dots ,S_e$.

Formal definition

Let $T$ be a regular chain of $\mathbf{k}[x_1,\dots ,x_n]$ for some ordering of the variables $\mathbf{x}=x_1,\dots ,x_n$ and a real closed field $\mathbf{k}$. Let $\mathbf{u}=u_1,\dots ,u_d$ and $\mathbf{y}=y_1,\dots ,y_{n-d}$ designate respectively the variables of $\mathbf{x}$ that are free and algebraic with respect to $T$. Let $P\subset \mathbf{k}[\mathbf{x}]$ be finite such that each polynomial in $P$ is regular w.r.t.\ the saturated ideal of $T$. Define $P_{>}:=\{p>0\mid p\in P\}$. Let $\mathcal{𝒬}$ be a quantifier-free formula of $\mathbf{k}[\mathbf{x}]$ involving only the variables of $\mathbf{u}$. We say that $R:=[\mathcal{𝒬},T,P_{>}]$ is a regular semi-algebraic system if the following three conditions hold.

• $\mathcal{𝒬}$ defines a non-empty open semi-algebraic set $S$ of ${\mathbf{k}}^d$,
• 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 $Z_{\mathbf{k}}(R)$, is defined as the set of points $(u,y)\in {\mathbf{k}}^d\times {\mathbf{k}}^{n-d}$ such that $\mathcal{𝒬}(u)$ is true and $t(u,y)=0$, $p(u,y)>0$, for all $t\in T$and all $p\in P$.

See also

• Characteristic set
• Triangular decomposition
• Regular chain
• RegularChains
• Real algebraic geometry

References

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