Let $V$ be a real affine variety in $\mathbb R^n$, i.e. the zero set of a real polynomial $p(x\_1,\dots,x\_n)$. Consider the following three definitions of the dimension of $V$, $dim(V)$.



> 
> ***Definition 1***: if $I$ is the ideal of
>  polynomials vanishing on $V$, then
>  $dim(V)$ is the maximum dimension of a
>  coordinate subspace in $V(\langle
> > LT(I)\rangle)$ with a given graded
>  order $>$ on the monomials
>  ($\sum\alpha\_i>\sum\beta\_i$ implies
>  $x^\alpha>x^\beta$) (see the book
>  "Ideals, Varieties, and Algorithms"
>  chapter 9)
> 
> 
> 


***Definition 2***: $dim(V)$ is the largest $d$ such that there exists an injective semi-algebraic map from $(0, 1)^d$ to $V$ (see "Algorithms in Real Algebraic Geometry" chapter 5)



> 
> ***Definition 3***: $dim(V)$ is the largest $d$ such that there exists a
>  subset of $V$ homeomorphic to $(0,
> > 1)^d$
> 
> 
> 


Are these three definitions, for the case of real affine varieties, pairwise equivalent, or pairwise different, or something else?



dimension of a real affine variety
They are all equivalent, including definition 1, to the Krull dimension of $S/I$, where $S=\mathbb{R}[x\_1,\ldots,x\_n]$ is the polynomial ring $I$ lives in. This is very good news for people like me who want to apply algebraic geometry to statistics, where numbers are mostly real.


Here's how it goes: 


Definition 1 is a way of computing the Krull dimension of $S/I$ via Groebner bases. See, e.g., p. 250 of *Computing in algebraic geometry: a quick start using SINGULAR* by Wolfram Decker and Christoph Lossen


Definition 2 is shown equivalent to Krull dimension in Corollary 2.8.9 of *Real Algebraic Geometry* by Bochnak, Coste and Roy. Note that they define the "dimension" of a real variety to be Krull dimension of its coordinate ring. I recommend reading the whole chapter.


Definition 3 is equivalent to Defintion 2 because any semialgebraic set admits a decomposition into finitely many pieces homeomorphic to $(0,1)^d$ (see Theorem 2.3.6 in RAG), and a finite union of semialgebraic sets of dimension less than $d$ cannot contain a set of dimension $d$. I.e., the only way a real variety can contain an open set homeomorphic to $(0,1)^d$ is by containing a semialgebraic set homeomorphic to $(0,1)^d$ in its semialgebraic cell decomposition.


