6 problems

Segre–Bartocci conjecture. The positive integers $t$ for which $D(x^t)$ is a hyperoval in $\mathbb{P}^2(\mathbb{F}_{2^k})$ for infinitely many $k$ are exactly $t=6$ and $t=2^i$ wit…

Let $PG(2,2^d)$ be the projective plane over ${\mathbb F}_{2^d}$. A monomial hyperoval is a hyperoval projectively equivalent to … The known examples include the regular hyperoval…

A projective plane of even order is a finite projective plane in which every line contains an even number of points. A hyperoval is a set of $q+2$ points in a projective plane of o…

A translation plane is a finite projective plane containing a group of elations acting transitively on its affine points; a translation hyperoval is a hyperoval invariant under thi…

Dempwolff's conjecture. $n$-dimensional doubly dual hyperovals over $\mathbb{F}_2$ exist only if $n$ is odd.

Let $k$ be even, and let $g_k(x,y,z)$ be the projective plane curve associated with the monomial hyperoval construction. An absolutely irreducible factor over $\mathbb{F}_2$ means…