13 problems

Let $q=p^n$ be a prime power with characteristic $p>3$, and let $X^d$ be a monomial over $\mathbb{F}_q$. A monomial is planar if its associated function has the planar property. De…

Let $r$ be a positive integer, let $t$ be an even positive integer, and let $a$ be a nonzero element of $\field_{2^r}$. A function $c\mapsto ac^t$ is called planar on a field of ch…

Let $q=p^n$, and write $s_p(m)$ for the sum of the base-$p$ digits of the positive integer $m$. Let $e\star d$ denote the operation used in the source on the exponents modulo $q-1$…

Let $p$ be an odd prime and $n$ a positive integer. A polynomial $f\in\mathbb{F}_{p^n}[X]$ is planar if, for every $a\in\mathbb{F}_{p^n}\setminus\{0\}$, the polynomial…

Monomial planar-function conjecture. These are the only monomial planar functions.

Let $f\colon \mathbb{F}_{2^n}\to\mathbb{F}_{2^n}$ be a planar function, and suppose that $f$ is a monomial. The three known families are the trivial monomials $f(x)=cx^{2^m}$, the…

Existence conjecture. There should exist an O'Nan configuration containing the point $(\infty)$ in each of these unitals.

Let $p$ be an odd prime, let $m$ be a positive integer, and consider planar monomial mappings $x\mapsto x^d$ on $\mathbb{F}_{p^m}$ with $d<p^m$ and $p\nmid d$. The known examples i…

Let $n$ be a positive integer, let $t$ satisfy $1\leq t\leq 2^n-2$, and let $c\in\mathbb{F}_{2^n}^*$. A monomial $x\mapsto cx^t$ is called planar when, for every…

Let $p$ be prime, let $r$ be a positive integer, and consider monomials $x^t$ over $\mathbb{F}_{p^r}$ with $p\nmid t$ and $t<p^r$. The known examples are $t=p^i+1$ when $0\le i<r$…

Let $p$ be a prime, let $n$ range over positive integers, and consider the monomial function $x^t$ on $\mathbb{F}_{p^n}$. Conjecture PN2. If $x^t$ is a planar function on…

Let $p$ be a prime, let $q=p^n$, and consider monomial functions $f(x)=x^t$ from $\mathbb{F}_q$ to $\mathbb{F}_q$. The known planar monomial exponents are those listed in Table 1 o…

Conjecture PN3. The polynomial $g_t(x,y)$ has an absolutely irreducible factor defined over $\mathbb{F}_p$ for every $t$ not of the form $p^i+1$ when $p\geq 3$, or $(3^i+1)/2$ when…