Timeline for Estimating the size of set of primes $p$ for which the polynomial is bijective in $\mathbb{F}_p[X]$
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 8, 2021 at 9:39 | comment | added | Peter Mueller | @Hhhhhhhhhhh No need to invoke Deligne here. If the monodromy group of $f$ over $\mathbb C$ is $S_n$, then $F(X,Y)=(f(X)-f(Y))/(X-Y)$ is absolutely irreducible over $\overline{\mathbb Q}$. A Hilbert Nullstellensatz argument shows that $F(X,Y)$ is absolutely irreducible over $\mathbb F_p$ for all sufficiently big primes $p$. Then Weil (the way easier $1$-dimensional precursor of Deligne) shows that $F=0$ has $p+O(\sqrt{p})$ points over $\mathbb F_p$. In particular, $f$ won't be injective. All this is well known, see e.g. the reference to Turnwald in David E Speyer's answer. | |
| Nov 8, 2021 at 7:32 | comment | added | Hhhhhhhhhhh | Could you please give reference to the result of Deligne that you are using here.Thanks | |
| Oct 29, 2021 at 15:15 | history | edited | Will Sawin | CC BY-SA 4.0 |
added 10 characters in body
|
| Oct 29, 2021 at 14:54 | history | answered | Will Sawin | CC BY-SA 4.0 |