In Wiki's page on Chevalley–Warning theorem, under "Statement of the theorems", it's written that
Chevalley–Warning theorem states that [...] the cardinality of the vanishing set of ${\displaystyle \{f_{j}\}_{j=1}^{r}}$ [...].
What does "the cardinality of the vanishing set of ${\displaystyle \{f_{j}\}_{j=1}^{r}}$" mean? (What is a vanishing set of polynomials? What is its cardinality?)
On
As far as I understand, a vanishing point of a polynomial is a point in which the polynomial vanishes, i.e. equals $0$. Thus vanishing point is just a fancy name for what that also known as a "root" or a "solution" of the polynomial.
A vanishing set of a polynomial is therefore the polynomial's set of roots/solutions. Its cardinality is the number of roots.
This is given by interpreting the multivariate polynomials $f_i(x_1,x_2,..x_m)$ as functions from $\mathbb{F}^m_q$ to $\mathbb{F}_q$, by the evaluation rule \begin{equation} (\alpha_1,\alpha_2,...\alpha_m)\mapsto f_i(\alpha_1,..,\alpha_m) \end{equation}
Where $\mathbb{F}^m_q$ here is just $m$ tuples of elements of $\mathbb{F}_q$.
So in this sense, we can talk about the tuples $(\alpha_1,..,\alpha_m)$ for which all these maps equal zero. This is our vanishing set, and its a subset of the (finite) set $\mathbb{F}^m_q$. The cardinality of this set is just the number of elements in this finite set.
As an example, take $f(x,y,z)=x^2+y^2+z^2$ over the field $\mathbb{F}_2$. We have solutions $(1,0,1),(1,1,0),(0,1,1)$ and $(0,0,0)$, so we have $4$ solutions, so the solution set has cardinality $4$. This is an instance of the Chevalley warning theorem in action and would be good to work through the proof for to gain understanding.