A note on the Chevalley–Warning theorems

Let $f_1,\dots,f_r$ be polynomials in $n$ variables, over the field $\mathbb{F}_q$, and suppose that their degrees are $d_1,\dots,d_r$. It was shown by Warning in 1935 that if $\mathscr N$ is the number of common zeros of the polynomials $f_i$, then $\mathscr N\geqslant q^{n-d}$. It is the main aim of the present paper to improve on this bound. When the set of common zeros does not form an affine linear subspace in $\mathbb{F}_q^n$, it is shown for example that $\mathscr N\geqslant2q^{n-d}$ if $q\geqslant4$, and that $\mathscr N\geqslant q^{n+1-d}/(n+2-d)$ if the $f_i$ are all homogeneous.
Bibliography: 5 titles.