8 problems

Nonexistence conjecture. The polynomial $f(x)$ is not a permutation polynomial over $\mathbb{F}_{q^2}$.

Five-valued conjecture. If either $d_1+d_2\geq 5$, or $d_1+d_2=3$ and $p^n\equiv 11\pmod{12}$, then the Weil spectrum over $L$ is at least five-valued. In the first case its five v…

Let $F$) be a finite field of order $4^m$, where $m$ is even, and set … Let $W_{F,d}(a)$ denote the Weil sum associated with this decimation. Niho's five-value conjecture. The set…

Helleseth's vanishing conjecture. There is some $a\in F^*$ such that

Helleseth's three-valued conjecture. If $[F:{\mathbb F}_p]$ is a power of $2$, then there is no invertible exponent $d$ over $F$ such that $W_{F,d}$ is three-valued.

Symmetry conjecture. $W_{F,d}$ is symmetric.

Katz–Langevin–Lee–Sapozhnikov conjecture.

Let $F$ be a finite field of order $q=3^n$, where $n$ is odd and $n>1$, and let $d=3^r+2$ satisfy … For $a\in F$, define the Weil sum … where $\psi\colon F\to\mathbb C$ is the cano…