212 problems

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

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…

Let $K$ be a number field, let $S$ be a finite set of places of $K$ containing all archimedean places, and let $f:\mathbb{P}^1\to\mathbb{P}^1$ be a rational map of degree at least…

Let $(ft){t einLambda}$ be a non-isotrivial algebraic family of rational maps with degree $dgeq 2$, parametrized by an algebraic curve $\Lambda$ over $\mathbb C$. A rational map is *…

The $4d-1$ bound conjecture. One can take

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…

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…

Let $N$ be an integer with $N\geq 4$, and let $f(z)\in\mathbb{Q}[z]$ be a quadratic polynomial. A point is of exact period $N$ if its forward orbit under $f$ has least period $N$.…

Dynamical Siegel conjecture. For any nonpreperiodic point $\beta\in\mathbb{P}^1_{\mathfrak{o}_K}(\overline{K})$, there are at most finitely many preperiodic points of $\varphi$ in…

Let $k$ be a number field, let $S$ be a finite set of places of $k$ containing the archimedean places, and let ${\mathcal O}_{k,S}$ be the ring of $S$-integers. Let $R(x)\in k(x)$…

Generalized Lang conjecture. The set of pre-periodic points contained in $Y$ is not Zariski-dense in $Y$.

Let $X$ be a proper algebraic variety over a finitely generated field $K$ of characteristic zero, and let $f:X\to X$ be a morphism. Suppose there is a subset $S\subseteq X(K)$ that…

Let $S$ be a smooth irreducible quasi-projective variety over $\mathbb{C}$, let $\Phi:S\times\mathbb{P}^N\to S\times\mathbb{P}^N$ be an algebraic family of endomorphisms of degree…

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

Second-iterate irreducibility conjecture. If $f_r^2(x)$ is irreducible over $K$, then $f_r$ is stable over $K$.

Stable-factorization conjecture. The polynomial $f_r$ is eventually stable over $\mathbb{Q}(i)$ with constant $C(f_r,0)=4$, and exactly one of the following cases holds:

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,

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 $f$ be the post-critically finite quadratic polynomial under consideration, and let $G_n(f)$ and $M_n(f)$ denote, respectively, its level-$n$ Galois group and even Markov group…

Let $b\in\frac{1}{2}\mathbb{Z}$ and let $\phi\in\mathbb{Z}[x]$ be separable and quadratic. Assume that the forward orbit $\{\phi^n(b):n=1,2,\ldots\}$ is infinite. Strong Dynamical…

Let $K=\mathbb{F}_p(t)$), let $T$ be the complete infinite rooted binary tree, and let $f_t=x^2+t\in K[x]$. Write $G_{f_t}(0)$ for the arboreal Galois group associated with the orb…

Let $f\in\mathbb{F}_q[x]$ be a monic quadratic polynomial, and say that two polynomials are conjugate when they are related by the relevant change of coordinates. Let…

Let $\phi\in\mathbb{Z}[x]$ be quadratic, with infinite critical orbit and all iterates irreducible. For an integer sequence $a_n=\phi(a_{n-1})$, let $P(a_n)$ denote the largest pri…

Let $S$ be a smooth irreducible quasi-projective variety over ${\mathbb C}$, let $K$ be a positive integer, and let … be a family of endomorphisms with $\Phi(s,p)=(s,F_s(p))$, wher…

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