  Let $R$ be a finite ring and define the paraboloid $P = \{(x_1, \dots,
x_d)\in R^d|x_d = x_1^2 + \dots + x_{d-1}^2\}.$ Suppose that for a sequence of
finite rings of size tending to infinity, the Fourier transform of $P$
satisfies a square-root law for some fixed constant $C$. Then all but finitely
many of the rings are fields. Most of our argument works in greater generality:
let $f$ be a polynomial with integer coefficients in $d-1$ variables, with a
fixed order of variable multiplications (so that it defines a function
$R^{d-1}\rightarrow R$ even when $R$ is noncommutative), and set $V_f = \{(x_1,
\dots, x_d)\in R^d|x_d = f(x_1, \dots, x_{d-1})\}$. If (for a sequence of
finite rings of size tending to infinity) we have a square root law for the
Fourier transform of $V_f$, then all but finitely many of the rings are fields
or matrix rings of small dimension. We also describe how our techniques let us
see that certain varieties do not satisfy a square root law even over finite
fields.
