I'm really not sure about this, but I've heard somewhere that for any prime $p$,
$|\sum_{x=0}^{p-1} (\frac{ax^3 +bx^2 +cx +d}{p} ) |\le \sqrt{2p}$ holds.
Does anyone know a proof for this inequality or a similar result?
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4$\begingroup$ It should be $2\sqrt{p}$. This is Hasse's theorem on elliptic curves. $\endgroup$– WojowuCommented Apr 1, 2020 at 12:00
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1$\begingroup$ At least for $a\neq 0$... For $a=c=d=0$, $b=1$, it is obviously false! $\endgroup$– abxCommented Apr 1, 2020 at 12:17
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$\begingroup$ Wow thanks!! That was what I was looking for $\endgroup$– JunsukimCommented Apr 1, 2020 at 13:15
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$\begingroup$ A nice self-contained proof is given in Voloch's answer to mathoverflow.net/questions/87916. $\endgroup$– Peter MuellerCommented Apr 1, 2020 at 16:03
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