= Solution
The <upper critical dimension> is the dimension at which the leading nonlinear interaction is marginal under Gaussian scale transformations. Normalize the gradient term and rescale $x=b x'$ so that $\phi(x)=b^{-(D-2)/2}\phi'(x')$. A coupling $g_{2p}\int\phi^{2p}\,d^Dx$ then scales by
$$
g'_{2p}=b^{y_{2p}}g_{2p},\qquad y_{2p}=D-p(D-2)=2p-(p-1)D.
$$
Thus
$$
\boxed{D_c=\frac{2p}{p-1}:\quad D_c=4\text{ for a quartic critical point},\quad D_c=3\text{ for a sextic tricritical point}.}
$$
Below this dimension, the interaction is a <relevant operator> at the <Gaussian fixed point>, so neglecting long-wavelength fluctuations is not self-consistent near criticality. A direct <Ginzburg criterion> gives the same result: the fluctuation in a correlation volume scales as $\langle(\delta\phi)^2\rangle_\xi\sim\xi^{2-D}$. For ordinary ordering $M^2\sim r/u\sim\xi^{-2}/u$, their ratio is proportional to $u\xi^{4-D}$. Along a tricritical trajectory $M^2\sim(|r|/v)^{1/2}\sim\xi^{-1}/\sqrt v$, giving ratio proportional to $\sqrt v\,\xi^{3-D}$. These ratios grow without bound below the corresponding <upper critical dimension>.
At $D=D_c$, the interaction is marginal. \b[Mean-field powers can survive, but acquire multiplicative logarithmic corrections]; it is too strong to say that all mean-field exponent values necessarily fail at equality. Above $D_c$, fluctuations do not change the leading <mean-field critical exponents>, although a <dangerously irrelevant coupling> can invalidate naive hyperscaling.
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