216 problems

Non-square conjecture. For every $n\ge3$, both $b_n^+(c)$ and $b_n^-(c)$ are non-square in $\mathbb{Z}[i]$.

For every integer $d\ge 2$ and every number field $K$, there exists a constant $\epsilon_{K,d}>0$ such that…

Let $K$ be a number field, let $f\in K(x)$ be a rational map of degree at least $2$, and let $\alpha\in\mathbb{P}^{1}(\overline{K})$ be non-exceptional, meaning that its backward o…

Let $K$ be a number field, let $f\in K(x)$ have degree at least $2$, and let $\alpha\in\mathbb{P}^{1}(K)$. Define the backward orbit…

Let $K$ be a global function field of characteristic $p$, let $f\in K[x]$ have degree $d\ge 2$, and let $a\in K$. For $n\ge 1$, let $K_n$ be the splitting field over $K$ of…

Dynamical Mordell–Lang conjecture. For every $x\in X(K)$, the return set is a finite union of arithmetic progressions. This conjecture is a central problem in arithmetic dynamics.…

Morton–Silverman conjecture. There exists a constant $C=C(d,N,[k:{\mathbb Q}])>0$ such that every degree $d$ endomorphism $f$ of ${\mathbb P}^N$ over $k$ satisfies

Kawaguchi–Silverman conjecture. The arithmetic degree at $x$ equals the dynamical degree of $f$:

Vojta's Main Conjecture. Given $\epsilon > 0$, there exists a Zariski-closed subset $Z=Z_\epsilon eq X$ such that

Let $X$ be an irreducible variety defined over an algebraically closed field $F$ of characteristic $0$, and let $\phi\colon X\dashrightarrow X$ be a dominant rational map. A domina…

Hutz's conjecture. There is no even degree $d>2$ and no $c\in\mathbb Q$ such that $\varphi_{d,c}$ has a rational point of exact period $n$. Moreover,

Let $N\geq 4$. A quadratic polynomial is a polynomial $f(x)\in\mathbb{Q}[x]$ of degree $2$, and a rational point of period $N$ is an $x\in\mathbb{Q}$ such that $f^{(N)}(x)=x$ but…

Flynn–Poonen–Schaefer conjecture. There is no quadratic polynomial $\varphi_{2,c}(z)=z^2+c\in\mathbb Q[z]$ with a rational point of exact period $n$.

Let $K$ be a number field, let $K^{\mathrm{ab}}$ be its maximal abelian extension, let $f:{\mathbb P}^1\to{\mathbb P}^1$ be a polynomial map of degree $d\geq2$ defined over $K$, an…

Let $k$ be an algebraically closed field, let $V$ be a variety over $k$, and let $(V,\phi)$ be a $\sigma$-variety over $k$, with $\phi$ the self-map defining its difference structu…

For an individual rational function $f\in\operatorname{Rat}_d(k)$, let $h_{\operatorname{crit},\mathbb{Q}}$ be the critical height on the moduli space $\mathrm{M}_d$, and let…

Let $\phi(x)=x^2+c$ for some $c\in\mathbb{Q}$, and let $a\in\mathbb{Q}$ be a periodic point for $\phi$, meaning that $\phi^n(a)=a$ for some $n\geq1$; the least such $n$ is the peri…

Zhang's conjecture. There exists a point $x\in X({\mathbf{k}})$ whose orbit $O_f(x)$ is Zariski dense in $X$. This is a polarized special case of the Zariski dense orbit problem an…

Let $K/\mathbb{Q}$ be a quadratic field, let $f\in K[z]$ be a quadratic polynomial, and let $\operatorname{PrePer}(f,K)$ denote its set of $K$-rational preperiodic points. Doyle's…

Let $f(x)=x^2-1$, and for an integer $n\geqslant 2$ let $P(n)$ denote the set of prime numbers arising in the associated sequence defined in the paper. Density-zero conjecture. For…

Let $S=\{x^2+c_1,\dots,x^2+c_r\}$ for some distinct $c_1,\dots,c_r\in\mathbb{Q}$ and $r\geq2$. Let $M_S$ denote the semigroup generated by $S$ under composition, and call a subset…

Let $K$ be a Hilbertian field of characteristic $0$, and let $n\geq 2$. For a polynomial $f\in K[x]$, write $f^{-k}(0)$ for the set of $k$-th preimages of $0$, and let $[S_n]^k$ de…

Let $f(z)=z^3+\frac{1}{c}$ with $c\in\mathbb{Z}\setminus\{0\}$. Here $f^n$ denotes the $n$-fold iterate of $f$. Eventual stability conjecture. If $f(z)$ is reducible over…

Adelic Zariski dense orbit conjecture. The stated adelic-open-set conclusion holds.

Let $X$ be a variety and let $\Phi:X\to X$ be a dominant map. A point of $X$ is preperiodic for $\Phi$ if its forward orbit under $\Phi$ is finite, and a subvariety $Y$ is preperio…