We study the relationship between two measures of Boolean functions;
\emph{algebraic thickness} and \emph{normality}. For a function f, the
algebraic thickness is a variant of the \emph{sparsity}, the number of nonzero
coefficients in the unique GF(2) polynomial representing f, and the normality
is the largest dimension of an affine subspace on which f is constant. We
show that for 0<ϵ<2, any function with algebraic thickness
n3−ϵ is constant on some affine subspace of dimension
Ω(n2ϵ). Furthermore, we give an algorithm
for finding such a subspace. We show that this is at most a factor of
Θ(n) from the best guaranteed, and when restricted to the
technique used, is at most a factor of Θ(logn) from the best
guaranteed. We also show that a concrete function, majority, has algebraic
thickness Ω(2n1/6).Comment: Final version published in FCT'201