Question

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?

Answer 1

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.

Answer 2

At least for definitions 2 and 3 : ${\rm dim}_2 (V) \leq {\rm dim}_3 (V)$ since a semialgebraic map is peicewise a homeomorphism.

Answer 3

Definition 2 and 3 are equivalent. What you want here is the notion of Cylindrical Algebraic Decomposition (see e.g. Bochnak, Coste and Roy's book). The $d$ you're looking for in both cases is the dimension of the largest cell in the decomposition, and it is a routine result that this dimension does not depend on the decomposition itself.

As for Definition 1, I am not sure I understand exactly what you're saying. You're using a total-degree monomial ordering, then taking the zero-set of the leading terms of the variety's ideal... I'll come back to this if I remember.