100 problems

Let $P$ be a complex polynomial of degree $n\ge2$ satisfying $P(0)=0$ and $P'(0)=1$. A point $\zeta$ is a critical point of $P$ when $P'(\zeta)=0$. Smale's mean value conjecture. T…

Let $K$ be a global field of characteristic not $2$, let $f\in K[x]$ be a quadratic polynomial, and let $\alpha\in K$. Suppose that the backward orbit … contains no ramification po…

Markov group containment conjecture. For every $n\geq 1$, the associated Markov group is contained in the corresponding Galois group:

Let $\mathcal{C}_d$ be the connectedness locus of the space $\mathcal{P}_d$ of degree-$d$ polynomials, and let a polynomial in $\mathcal{C}_d$ be combinatorially rigid if it has no…

Let $P$ be a complex polynomial of degree $n\ge2$ satisfying $P(0)=0$ and $P'(0)=1$. A point $\zeta$ is a critical point of $P$ when $P'(\zeta)=0$. Dual Smale's mean value conjectu…

Let $A\in\mathbb{F}_2[x]\setminus\{0\}$. Define the associated sequences $(a_{2k})_k$, $(A_{2k})_k$, and $(A_{2k+1})_k$ by $A_0=A$, $A_{2k}=1+(x^2+x+1)A_{2k-1}$, and by removing al…

Let $P$ be a polynomial having a parabolic cycle, and let $J(P)$ denote its Julia set. A small perturbation of $P$ should convert the immediate basin of the parabolic cycle into an…

Let an ODE system be given by polynomial equations of the form described above, and let a monomial quadratization mean a quadratization obtained by introducing monomial functions o…

Let $f_c(z)=z^2+c$. For positive integers $m,n$ with $m\mid n$, define the delta factors $\Delta_{n,m}$ by … and, when $m=n$, by … Here $\Phi^{\mathrm{cyc}}_k$ is the $k$-th cyclot…

Let $a\in\mathbb{Z}$ with $a\notin\{0,-1\}$. The rational map $\phi(z)=z^2+1/a$ has degree two, and a pair $(\phi,\alpha)$ is eventually stable when the number of irreducible facto…

Hinkkanen–Martin's modified conjecture. For all $f,g\in G$, one has

Folklore conjecture. The map $f$ can be approximated by polynomial maps with positive entropy and by polynomial maps with finitely many periodic orbits.

A complex quadratic is a polynomial of the form $z\mapsto z^2+c$. A polynomial is non-hyperbolic when it is not hyperbolic, and it is topologically rigid when it is not topological…

A reduced mapping schema $S$ has a connectedness locus $\mathcal{C}^S$. For two reduced mapping schemata $S$ and $S'$, write $S\succ S'$ when $\mathcal{C}^S$ contains a hyperbolic…

Let $f$colon $\mathbf{C}\to\mathbf{C}$ be a monic, centered polynomial of degree $d\geq 2$, and let $\mathcal{P}^{d-1}$ be the affine space of such maps. Let…

Let $N\geq 3$, let $A_j(t)$, for $j=2,3,\ldots,N-1$, be polynomial functions, and consider the differential equation … A solution is a center at the origin if nearby solutions have…

Let $f\in\mathbb{Z}[x]$ be a quadratic polynomial. Call $f$ stable if every iterate $f^n$ is irreducible over $\mathbb{Q}$, and call $f$ critically infinite if the forward orbit of…

Let $f$ be a polynomial of degree $d$ with an indifferent fixed point at the origin. Let $c_i$ be the critical points of $f$, let $\theta$ be the rotation number at the origin, and…

Let $F$ be a field and let $f in F[x]$ be a post-critically finite polynomial. Two polynomials are equivalent when they are related by the equivalence relation used in the source,…

Let $f(x)\in\mathbb{Z}[x]$ have degree greater than $1$, and consider the partial products of … A continued fraction is specializable if its partial quotients specialize appropriat…

Let $k$ be an algebraically closed complete non-Archimedean field, possibly of positive characteristic. Let $\mathcal{J}$ be the Julia set of a polynomial of degree at least $2$ th…

Polynomial-degree conjecture. The number of potential localization regions for Hopf and Bogdanov–Takens bifurcations grows as $n-2$.

Hyperbolic polynomial branch and extension conjecture. The following are equivalent:

Let $f$ and $\widetilde f$ be polynomials, and suppose they are combinatorially equivalent under an appropriate definition of combinatorial equivalence. Combinatorial Rigidity Conj…

Let $F=F_{a,v}$ be a cubic polynomial in the superattracting period-$p$ parameter curve $4\mathcal{S}_p4$, written in the normal form … Assume that $F$ is a Misiurewicz map, meanin…