# An Algebraic Approach to Identifiability
## 2 Preliminaries
### Ideals and Varieties  
The succeeding sections will make use of polynomial ideals and their corresponding varieties. Therefore, the required concepts used herein are briefly introduced as well. Consider the commutative multivariate polynomial ring \(\mathbb{R}[x_{1},\ldots,x_{n}]=\mathbb{R}[x]\) in the variables \(x_{1},\ldots,x_{n}\). An ideal \(I\subseteq\mathbb{R}[x]\) is a special subset of this ring satisfying the conditions  
1. \(0\in I\)
2. \(a,b\in I\implies a+b\in I\)
3. \(a\in I,c\in\mathbb{R}[x]\implies ca\in I\). Thus, an ideals contains an infinite number of polynomials, let alone the trivial ideal \(\{0\}\). Nonetheless, [PERSON]'s basis theorem asserts that these ideals can be represented using a finite polynomial set \(G=\{g_{1},\ldots,g_{s}\}\subset\mathbb{R}[x]\), a so-called basis, by  
\[I=\langle g_{1},\ldots,g_{s}\rangle=\{a_{1}g_{1}+\cdots+a_{s}g_{s}\,|\,a_{1}, \ldots,a_{s}\in\mathbb{R}[x]\}.\]  
There are different bases that represent the same ideal, possibly with a different number of generators. Ideals can be compared by computing a so-called standard basis or _Grobner basis_. A reduced Grobner basis is unique for a given monomial ordering, and there is an algorithm to compute them. On the other hand, one can easily verify whether a polynomial is contained in an ideal given a Grobner basis of that ideal. This allows deciding subset relations of ideals (and thus also their equality). The common real zero set \(V=\mathbf{var}^{\mathbb{R}}(I)\subseteq\mathbb{R}^{n}\) of all polynomials in an ideal \(I\subseteq\mathbb{R}[x]\) is called the _real variety_ of that ideal. This mapping from ideals to varieties is not bijective. Consider, for example, the ideals \(\langle x\rangle,\langle x^{2}\rangle\subseteq\mathbb{R}[x]\) over the univariate ring, which are different, but whose real varieties \(\{0\}\subseteq\mathbb{R}\) coincide. Given a real variety \(V\subseteq\mathbb{R}^{n}\), there is, however, an ideal \(\operatorname{Id}(V)\subseteq\mathbb{R}[x]\) containing all polynomials that vanish on \(V\). Such an ideal is called _real_, and there is a one-to-one correspondence between real ideals and real varieties. This real ideal is the superset of all ideals whose polynomials evaluate to zero on the real variety and is called the _real radical_ of that ideals. The real radical \(\mathbf{rad}^{\mathbb{R}}(I)\) of an ideal \(I\) can also be computed algebraically (29, p. 85). One can also consider the set of polynomials in \(\mathbb{R}[x]\) that vanish on a set \(W\subseteq\mathbb{R}^{n}\), which is not a real variety. These polynomials will then have additional zeros  
\[W\subseteq\mathbf{var}^{\mathbb{R}}(\operatorname{Id}(W))=\overline{W} \subseteq\mathbb{R}^{n}.\]  
This variety \(\overline{W}\) is called the _Zariski closure_ of \(W\), the smallest variety containing \(W\). Like the real radical, there is a correspondence of geometric set operations on the varieties and operations on the ideals, which make it possible to handle these operations algebraically. The intersection \(V\cap W\) of real varieties corresponds to the _sum_\(I+J\) of the ideals \(I\) and \(J\):  
\[\mathbf{var}^{\mathbb{R}}(I+J)=\mathbf{var}^{\mathbb{R}}(I)\cap\mathbf{var}^ {\mathbb{R}}(J).\]  
The sum is defined as the set of all polynomials contained in \(I\) or \(J\) and is also an ideal. For the union of varieties, one has  
\[\mathbf{var}^{\mathbb{R}}(I\cap J)=\mathbf{var}^{\mathbb{R}}(I)\cup\mathbf{ var}^{\mathbb{R}}(J),\]  
where  
\[I\cap J=\{g\,|\,g\in I\wedge g\in J\}\]  
is the _intersection_ of the ideals. The difference set \(V\setminus W\) of two varieties \(V\) and \(W\) is, in general, not a variety anymore. As an example, consider in \(\mathbb{R}^{1}\) the varieties \(V=\mathbb{R}\) and \(W=\{0\}\) with corresponding real ideals \(\langle 0\rangle\) and \(\langle x\rangle\), respectively. There is no ideal with real variety \(V\setminus W\).