25 problems

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…

Corrected Ruelle zeta-function conjecture. For any $x_1,\dots,x_k\in\mathbb F_q\setminus\{0,1,t\}$ with $k\ge 1$, the series

Let $f$ be a rational map, and let its algebraic entropy be the exponential growth rate of the degrees of its iterates. Bellon–Viallet conjecture. The algebraic entropy of every ra…

Zariski-dense orbit conjecture. There exists $x\in X(k)$ whose forward orbit under $f$ is well-defined and Zariski-dense in $X$.

Let $k$ be a field, let $n$ be a positive integer, and let $f\in\operatorname{End}(\mathbb{A}^n_k)$. Its dynamical degree is … where $\deg(f)$ is the maximum of the degrees of the…

Let $T$ be an irreducible variety, let $X/T$ be a family of smooth irreducible projective varieties, and let $f:X/T\dashrightarrow X/T$ be a family of dominant rational maps. Write…

Let $f:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ be a dominant rational map, and let … be its dynamical degree. Bellon–Viallet conjecture. The dynamical degree is an algebraic integ…

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

Let $X$ be a normal projective variety and let $f:X\to X$ be a non-isomorphic surjective endomorphism. The properties weakly $\delta$-imprimitive, strongly imprimitive, quasi-abeli…

Let $X$ be a projective variety over an algebraically closed field $k$ of characteristic zero, and let $f:X\dasharrow X$ be a dominant rational self-map. Write $k(X)^f$ for the fie…

Let $X$ be a smooth projective variety over an algebraically closed field $K$ of characteristic $0$, and let $\phi:X\dashrightarrow X$ be a dominant rational self-map. Assume that…

Let $X$ be a smooth projective variety defined over an algebraically closed field $K$ of characteristic $0$, and let $\phi:X\dashrightarrow X$ be a birational self-map. Its first d…

Let $X:=\mathbb{P}^n$, let $f:X\to X$ be an endomorphism with $\deg(f)\geq 2$, and let $Y$ be a closed subvariety of $X$. The linear-subspace conjecture. If $Y$ is $f^{-1}$-invaria…

Let $K$ be a field of characteristic $0$, let $d$ be a positive integer, and let $f\colon \mathbb{A}^d_K\to\mathbb{A}^d_K$ be a polynomial automorphism. Blanc–Van Santen's conjectu…

Let $K$ be a field of characteristic $0$, let $d$ be a positive integer, and let $f\colon \mathbb{A}^d_K\to\mathbb{A}^d_K$ be a polynomial map. Recall that a real number…

Sparsity conjecture. Let $d\ge3$ and $n\ge2$. For every fixed tail-length $\ell\ge1$, the set

Let $X$ be a smooth projective variety defined over an algebraically closed field $\mathbf{k}$, and let $f\colon X \vdash X$ be a dominant self-correspondence. For a prime $\ell$ d…

Let $\Gamma$ be a countable group and let $[\,\cdot\,]\colon\Gamma\to\mathbb{Z}$ be a homomorphism. Suppose that … where $M>\sum_{s\in I}f_s$, and $[s]\geqslant1$ and $f_s>0$ for e…

Uniform dynamical-degree lower-bound conjecture. There exists a constant $\gamma_N>0$ such that, for all dominant rational maps $f:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$,

Let $X$ be a quasi-projective variety over $\mathbb C$, let $F:X\to X$ be a finite endomorphism, and let $\{b_i\}_{i\geq 0}$ be points of $X(\mathbb C)$ satisfying … Let $V$ be a p…

Let $X$ be a quasi-projective variety defined over $\mathbb C$, let $f:X\rightarrow X$ be an endomorphism, and let $V$ be any subvariety of $X$. For a point $x\in X(\mathbb C)$, de…

Positive-characteristic dense-orbit conjecture. There exists a point $x\in X(k)$ whose orbit is Zariski dense in $X$.

Let $K$ be an algebraically closed complete nonarchimedean field, possibly with trivial absolute value, and let $X$ be an irreducible projective variety over $K$. Suppose…

Polynomial correction conjecture. The infimum $\ell_\varphi$ exists and is an integer satisfying $0\leq\ell_\varphi\leq N$.

Let $X$ be a projective variety over $\mathbb{C}$ and let $\varphi:X\to X$ be an endomorphism with $\varphi^*L\cong L^{\otimes q}$ for some ample line bundle $L$ on $X$. Suppose th…