  We study the structure of the Fourier coefficients of low degree multivariate
polynomials over finite fields. We consider three properties: (i) the number of
nonzero Fourier coefficients; (ii) the sum of the absolute value of the Fourier
coefficients; and (iii) the size of the linear subspace spanned by the nonzero
Fourier coefficients. For quadratic polynomials, tight relations are known
between all three quantities. In this work, we extend this relation to higher
degree polynomials. Specifically, for degree $d$ polynomials, we show that the
three quantities are equivalent up to factors exponential in $d$.
