We present three results regarding problems related to multivariate 
polynomials of low degree over a p-element finite field $F$:

1) Explicit construction of a seedless disperser for affine subspaces of $F^n$ 
of dimension greater than $2n/5 + 10$.

2) Explicit construction of a family of functions $f$ mapping $F^n$ to $F$ that 
have exponentially small correlation with degree-d multivariate polynomials 
over $F$.

3) An improved “global” list-decoding algorithm for Reed-Muller codes.

The common theme underlying these results is their method of proof. We 
translate the problems into questions about univariate polynomials over 
$F_{p^n}$. Functions represented by low-degree multivariate polynomials 
in $F[x_1, ... , x_n]$, when viewed as univariate polynomials over 
$F_{p^n}$, have a very special structure. We use various properties of this 
special structure to obtain the above mentioned results. 

Joint work with Swastik Kopparty