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Let $ \mathbb{F}_p$ be a finite field and $\{f_j\}_{j=1}^{j=r} \subseteq \mathbb{F}_p[X_1,...,X_n]$ be a set of polynomials.

Let consider the system of equations:

$f_j(x_1,...,x_n)=0$ for $j = 1,...,r$

We are interested in $N_p$ - the number of common solutions of this system modulo $p$.

By Chevalley–Warning theorem we know that if $n > \sum d_j$, where $d_j$ is the total degree of $f_j$ then $N_p = 0$. Of couse it is not criterion.

Whether there are results that connect $N_p$ with something subtler then $\sum d_j$?

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    $\begingroup$ See for instance the introduction to arxiv.org/abs/1408.3224 $\endgroup$
    – Laurent Berger
    Commented Jan 27, 2015 at 12:56
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    $\begingroup$ If by "the number of common solutions of this system modulo p" you mean "the number of common solutions of this system", taken modulo p, then Fulton's trace formula answers your question (see theorem 2 of math.stanford.edu/~dlitt/exposnotes/fultontrace.pdf) $\endgroup$
    – js21
    Commented Jan 27, 2015 at 13:13
  • $\begingroup$ @js21 Does Fulton's trace formula work even if the corresponding variety is not irreducible or non-smooth? $\endgroup$
    – Alexey Milovanov
    Commented May 1, 2016 at 7:18
  • $\begingroup$ By Chevalley-Warning theorem, $N_p \equiv 0 \pmod{p}$, and not what you wrote. In particular, if there is a solution, then there are at least $p$ of them. When $n > \sum d_j$, then there is also a nicer bound called Warning's second theorem which says that if there is a common solution, then there are at least $p^{n - \sum d_i}$ common solutions: arxiv.org/abs/1404.7793. $\endgroup$
    – Anurag
    Commented May 20, 2016 at 9:49

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