Let F be a field of positive characteristic $p$ and let $d,n$ be two positive integers.

Can we explicitly write down an equation defining a *smooth* hypersurface $X_d⊂\mathbb P^n_F$ of degree d ?

This is easy if $p$ does not divide $d$: the Fermat equation $∑x^d_i=0$ will do.

But what if $p$ *does* divide $d$ ?

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