Timeline for Known estimate for gaussian sum $\sum_{x \in \mathbb{F}_q} \psi( a x^m + b x^n)$?

7 events

when toggle format	what		by	license	comment
May 20, 2022 at 23:38	vote	accept	José
May 17, 2022 at 1:01	answer	added	Mark Lewko		timeline score: 3
May 16, 2022 at 23:19	comment	added	Ofir Gorodetsky		Weil proved in general that the modulus of $\sum_{x \in \mathbb{F}_q} \psi(f(x))$ is at most $(\deg f - 1)\sqrt{q}$ for any polynomial $f$ not of the shape $g^p-g$.
May 16, 2022 at 23:10	history	edited	José	CC BY-SA 4.0	added 11 characters in body
May 16, 2022 at 23:10	comment	added	José		Thank you. However, now I realize that I must also assume $m\ne n$, otherwise it would be just a monomial gaussian sum. I will edit the question to add this.
May 16, 2022 at 23:03	comment	added	LSpice		For $m = n = 2$ and $q$ odd, it is (what I think of as) a classical Gauss sum, whose value is well known: the square is $q\operatorname{sgn}_{\mathbb F_q}(-(a + b))$, and the sign is known in terms of how $\psi$ differs from the "basic" character $x \mapsto e^{2\pi i\operatorname{tr}_{\mathbb F_q/\mathbb F_p}(x)/p}$.
May 16, 2022 at 22:59	history	asked	José	CC BY-SA 4.0