  In this paper, two structural results concerning low degree polynomials over
finite fields are given. The first states that over any finite field
$\mathbb{F}$, for any polynomial $f$ on $n$ variables with degree $d \le
\log(n)/10$, there exists a subspace of $\mathbb{F}^n$ with dimension $\Omega(d
\cdot n^{1/(d-1)})$ on which $f$ is constant. This result is shown to be tight.
Stated differently, a degree $d$ polynomial cannot compute an affine disperser
for dimension smaller than $\Omega(d \cdot n^{1/(d-1)})$. Using a recursive
argument, we obtain our second structural result, showing that any degree $d$
polynomial $f$ induces a partition of $F^n$ to affine subspaces of dimension
$\Omega(n^{1/(d-1)!})$, such that $f$ is constant on each part.
  We extend both structural results to more than one polynomial. We further
prove an analog of the first structural result to sparse polynomials (with no
restriction on the degree) and to functions that are close to low degree
polynomials. We also consider the algorithmic aspect of the two structural
results.
  Our structural results have various applications, two of which are:
  * Dvir [CC 2012] introduced the notion of extractors for varieties, and gave
explicit constructions of such extractors over large fields. We show that over
any finite field, any affine extractor is also an extractor for varieties with
related parameters. Our reduction also holds for dispersers, and we conclude
that Shaltiel's affine disperser [FOCS 2011] is a disperser for varieties over
$F_2$.
  * Ben-Sasson and Kopparty [SIAM J. C 2012] proved that any degree 3 affine
disperser over a prime field is also an affine extractor with related
parameters. Using our structural results, and based on the work of Kaufman and
Lovett [FOCS 2008] and Haramaty and Shpilka [STOC 2010], we generalize this
result to any constant degree.
