we refer the reader to [@HFT]. In section \[ss:frequency\_hhp\] and \[ss:hpinch\] we discuss the frequency function and its relation to homogeneous harmonic polynomials. Although much of this is known, the estimates of these sections are much more refined than those previously in the literature, and we will need these results. In particular in Theorem \[t:eff\_tan\_con\_uniq\_harm\] we will prove an effective tangent cone uniqueness statement, which will play an important role in our estimates. In section \[ss:almost\_conesplitting\] we revisit the idea of cone splitting, introduced in this context first in [@chnava]. The results of [@chnava] are based on contradiction arguments, and we again prove much more refined estimates. Sections \[ss:almost\_codim2\_invariant\] and \[ss:symmetric\_criticalpoints\] discuss the relationship of critical points to the symmetry of a harmonic function. Finally in section \[ss:volume\_estimates\_harmonic\] we prove Theorem \[t:main\_critical\] for harmonic functions.

Homogeneous harmonic polynomials {#ss:hhp}
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Let $D\subset {\mathbb{R}}^n$ be any domain, and denote for convenience ${\mathcal{H}}(D)$ the space of harmonic functions $u:D\to {\mathbb{R}}$, $u\in W^{1,2}(D)$. Most of the times, we will consider $B(0,1)$ as our domain, thus we define for simplicity $B=B(0,1)\subset {\mathbb{R}}^n$. We recall that a polynomial $P$ is said to be homogeneous of degree $d$ if $P(\lambda x)=\lambda^d P(x)$ for all $\lambda\in {\mathbb{R}}$ 