13 problems

Let $p$ be prime, let $g$ and $h$ range over the parameters defining the fixed-point counting function, and let $N_{, g \mathop\mathsf{ANY}, h \mathop\mathsf{ANY}}(p)$ denote the c…

Average auxiliary-count conjecture.

Average fixed-point conjectures.

Relative-primitivity scaling conjectures.

Two-cycle count conjectures.

Auxiliary solution-count conjectures.

Two-cycle counting conjectures.

Fixed-point counting conjectures.

Let $q$ be a prime power and let $\textbf{F}_{q^n}$ be a finite field. Let $g\in\textbf{F}_{q^n}$ satisfy the conditions in Theorem, and suppose that $n\geq\log q$. Large-order con…

Average two-cycle conjecture. For every $C>0$,

Two-cycle counting conjectures. The paper conjectures

Let $l\in\mathbb{N}$, set $q=2^l$, and let $\beta\in\mathbb{F}_{q^6}^*$. Suppose that the quadratic equation … has two solutions $u_1,u_2\in\mathbb{F}_{q^6}$. Menezes–Teske–Weng's…

Let $\mathbb F_q$ be a finite field of odd characteristic. The polynomials $h_1,h_2\in\mathbb F_{q^d}[X]$ are required to be coprime and have degree at most $2$. Stronger polynomia…