= Solution
The real <scalar field> action has spacetime translations and <Lorentz invariance>, together giving <Poincaré symmetry>. With a real scalar transformation law it also has parity and <time reversal>. A generic polynomial potential has no continuous internal <symmetry>. The additional field reversal $\phi\mapsto-\phi$ is a <symmetry> precisely when every nonzero odd-power coupling vanishes. The free massless special case also admits a constant shift of $\phi$; generic interactions remove it.
In mass units, $[\partial_\mu]=1$ and the dimensionless <action> requires $[\mathcal L]=D$. Normalizing the kinetic term fixes the classical <mass dimension> of the <scalar field>:
$$
2+2[\phi]=D,\qquad[\phi]=\frac{D-2}{2},\qquad[m]=1.
$$
For each interaction, $[\lambda_n]+n[\phi]=D$, so
$$
\boxed{[\lambda_n]=D-\frac{n(D-2)}2.}
$$
These are classical or engineering <scaling dimensions>; quantum corrections can change the scaling of operators and couplings. The scale transformation prescribed later has field weight one, which equals this canonical weight only at $D=4$.
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