= Finite-field Kakeya polynomial bound
{title2=$|A|\geq\binom{q+n-1}{n}\geq q^n/n!$}
For a <finite-field Kakeya set> $A\subseteq\mathbb F_q^n$, a smaller <cardinality> would give a nonzero <polynomial> of <total degree of a polynomial> at most $q-1$ vanishing on $A$, by the <dimension of a bounded-total-degree polynomial space>. Its restriction to each complete <affine line in a vector space> has $q$ <roots of a polynomial> and degree less than $q$, hence vanishes identically. The top <homogeneous polynomial> part then vanishes in every direction. A <polynomial> of degree less than $q$ in each variable cannot vanish everywhere on $\mathbb F_q^n$ unless it is zero, giving a contradiction. For the <prime field>, this is the <polynomial nonvanishing below the field size>.
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