A Note on the Chevalley--Warning Theorems

Let $f_1,\...,f_r$ be polynomials in $n$ variables over a finite field $F$ of cardinality $q$ and characteristic $p$. Let $f_i$ have total degree $d_i$ and define $d=d_1+\...+d_r$. Write $Z$ for the set of common zeros of the $f_i$, over the field $F$. Warning showed that #(Z\cap H_1)\equiv#(Z\cap H_2)\mod{p} for any two parallel affine hyperplanes $H_1,H_2$ in $F^n$. We prove that the same congruence holds to modulus $q$. Warning also proved that # Z\ge q^{n-d} providing that $Z$ is non-empty. We sharpen this inequality in various ways, assuming that $Z$ is not a linear subspace of $F^n$.