Are these definitions of dimension for a real algebraic variety all equivalent?

Question

I would like to believe that the following notions of dimension are all equivalent for an a real algebraic variety in $\mathbb{R}^n$.

1. Lower Minkowski dimension
2. Upper Minkowski dimension
3. Hausdorff dimension
4. real dimension

(I realize that the algebraic dimension might be strictly larger.)

Is this true? Assuming so, any references to the literature that would be appropriate to cite for this fact would be appreciated.

I know that real algebraic varieties (and even semi-algebraic sets) are triangulable. Is that already enough?

Answer 1

First, the algebraic dimension is potentially bigger, not potentially smaller. Think about $V(x^2+y^2)$, for instance - this is algebraic dimension 1 but real dimension zero.

Second, yes, these are all equivalent for real varieties (where we mean the set of solutions to a system of polynomial equations in $\Bbb R^n$). Any such variety is a semi-algebraic set, which admits a stratification in to a finite union of locally closed submanifolds (see for instance corollary 3.8 in Michel Coste's Introduction to Semialgebraic Geometry). Since all three take finite unions to maximums, it suffices to show the claim for a smoothly semialgebraically-embedded copy of $(0,1)^d\subset\Bbb R^n$.

This immediately shows that the definition of 4 is $d$. For Hausdorff dimension, the fact the embedding is a diffeomorphism gives dimension $d$. The upper box dimension is also $d$, see here. Since the Hausdorff dimension is at most the lower box dimension is at most the upper box dimension, this shows that all three quantities are $d$.