# How to write down an explicit equation of given degree yielding a smooth hypersurface in a projective space?

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 $$d$$ ?

• The table in section 3 of Poonen's www-math.mit.edu/~poonen/papers/projaut.pdf does this. Dec 29, 2020 at 19:06
• (Katz also faced this problem in web.math.princeton.edu/~nmk/baby16.pdf —- see Section 6.) Dec 29, 2020 at 21:19
• @SashaP : thank you for your comment Dec 29, 2020 at 21:31
• In the case $p\vert d$ the equation mentioned by @alpoge is due to Ofer Gabber and is $x_0^d+\sum _{i=0}^{n-1}x_ix_{i+1}^{d-1}=0$. This is much better than my answer, which I'm thus deleting. Dec 30, 2020 at 0:22
• @alpoge Ofer Gaber's example in your reference is much better than mine. Could you please explain in an answer, (which can then be upvoted) , since I have deleted my own answer . Dec 30, 2020 at 0:26