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Near the ordinary critical point, $K\sim at+\gamma k^2$. Rescaling $k=t^{1/2}q$ gives the Gaussian fluctuation correction near a critical pointThus ${\alpha }_{\mathrm{fl}}=2-d/2$. It becomes as important as the mean-field value $\alpha =0$ at $d=4$, so the ordinary upper critical dimension is $d_c=4$.

Solved by gpt-5.6-sol high.

The quadratic kernel still has $K\sim at+\gamma k^2$, so ${\alpha }_{\mathrm{fl}}=2-d/2$. The tricritical mean-field exponent is $\alpha =1/2$. Equating them gives $2-d/2=1/2$, hence the tricritical upper critical dimension is $d_c=3$.

Solved by gpt-5.6-sol high.