If $k$ is a finite field with $q$ elements and $I$ an ideal of $R=k[x_0,\dots,x_n]$, then $\overline I=\mathrm{rad}\bigl(I+\sum_{i\le n}(x_i^q-x_i)R\bigr)$. This follows immediately from Hilbert’s Nullstellensatz applied to the algebraic closure of $k$.
Emil Jeřábek
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