= Solution
The relevant boundary here is the <upper critical dimension>: above it the critical interactions become negligible at long distances and the ordinary <mean-field critical exponents> apply; at it there can be logarithmic corrections. This differs from the <lower critical dimension>, below which a particular kind of ordering may fail to exist.
Normalize the coefficient of $(\nabla\phi)^2$ in the <Landau-Ginzburg theory>. Under a change of length scale $x=bx'$, its invariance gives the <engineering dimension> $x_\phi=(d-2)/2$. The quartic interaction then scales as
$$
u'=b^{d-4x_\phi}u=b^{4-d}u.
$$
It is a <relevant operator> below four dimensions and an <irrelevant operator> above four. Thus \b[the ordinary scalar transition has upper critical dimension $d_c=4$]. At a <tricritical point>, where the quartic interaction is tuned away, the sextic coupling scales as $v'=b^{6-2d}v$, giving \b[the tricritical upper critical dimension $d_c=3$]. This is the <upper critical dimension of an even scalar interaction> applied to powers four and six.
The <Ginzburg criterion> makes the reason for failure of the <Landau approximation> quantitative. In the ordered quartic theory, $M^2=|r|/u$, $V''(M)=2|r|$ and $\xi^2=K/(2|r|)$. The fluctuation of the <order parameter> averaged over a <correlation volume> has the scale
$$
\left\langle(\delta\phi)^2\right\rangle_{\xi}
\asymp k_BT\int_{|q|\lesssim\xi^{-1}}\frac{d^dq}{(2\pi)^d}
\frac1{Kq^2+2|r|}
\asymp\frac{k_BT}{K}\xi^{2-d}.
$$
Compared with the squared mean <order parameter>, the <Ginzburg criterion> therefore gives
$$
\frac{\langle(\delta\phi)^2\rangle_{\xi}}{M^2}
\asymp\frac{k_BTu}{K^2}\xi^{4-d}.
$$
For $d<4$ this ratio grows without bound as the <correlation length> diverges. The assumption that fluctuations are small relative to the ordered value fails arbitrarily near the transition, and the interacting <renormalization-group fixed point> determines the <critical exponents>. For $d>4$ the ratio tends to zero. At $d=4$ scale counting is marginal, and the running quartic coupling produces logarithms.
Similarly, at the <tricritical point> $M^2\asymp\sqrt{K/v}\,\xi^{-1}$, so the <multicritical Ginzburg ratio> scales as $(k_BT\sqrt v/K^{3/2})\xi^{3-d}$. The <Landau-Ginzburg theory> remains a useful effective description below its <upper critical dimension>; it is its fluctuation-free exponent calculation that fails, not the possibility of including fluctuations in the theory.
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