8 problems

Let $f\in\mathbb R[x,y]$ be neither additive nor multiplicative. For a finite set $A\subset\mathbb R$, write $f(A,A)=\{f(a_1,a_2):a_1,a_2\in A\}$. Near-quadratic Elekes–Rónyai expa…

Elekes's conjecture. The cardinality $|f(A,B)|$ should be superlinear in $n$, unless $f$ has a very concrete special form. This conjecture initiated the study of expansion of polyn…

Elekes's expansion conjecture. The image of $f$ on an $n\times n$ Cartesian product has cardinality superlinear in $n$, unless $f$ has a very concrete special form. This conjecture…

Near-quadratic expansion conjecture. For every $\varepsilon>0$,

Let $G$ be a graph and let $p$ be a polynomial such that the expansion of $G$ is bounded by $p$. A cost assignment is a function … For integers $t\geq 1$ and $q\geq 0$, a set…

Let $g$ be a polynomial over the integers. For each integer $a$, define … A geometric progression of length $b$ is a sequence of $b$ terms with a common ratio. Geometric-progressio…

Let $g(x)$ be a polynomial over the rationals, and write $g(\mathbb{Q})$ for its range on rational inputs. An arithmetic progression is a finite sequence of the form $a+id$ with fi…

Let $g(x)$ be a polynomial over the rationals of degree $d$, and let $G$ be a complex multiplicative group of rank $r$. Uniform boundedness conjecture. There is a function $b(d,r)$…