12 problems

A lacunary polynomial is a polynomial with a fixed number of terms, while the degrees and coefficients are unrestricted. Let $f$ and $g$ be polynomials, and consider their composit…

Let $\mathcal{Q}$ and $R_{m,n}$ be as above. For $F\in R_{m,n}$, let $Z$ denote the innermost polynomial of $F$. The vanishing innermost-polynomial conjecture. If…

Let $\mathcal{Q}$ and $R_{m,n}$ be as above. For a polynomial $F\in R_{m,n}$, let $W_F$ be the $F$-generator constructed in the paper, and let $\mathbb{T}$ be the corresponding cla…

Composition conjecture for moments. The moment conditions imply the composition condition: there should exist polynomials $\widetilde F$, $\widetilde G$ and $W$, with $F'=f$,…

Let $f=F'$ and $g=G'$ be polynomials, and let $A=[a,b]$. The polynomial Abel equation … has a center on $A$ if every solution with sufficiently small initial value satisfies…

Let $q$ be a prime power, and let $k,l$ be positive integers. Write $c\mathcal{F}_k(q)$ for the set of elements of degree $k$ over $c\mathbb{F}_q$. Existence conjecture. There exis…

Moment and Melnikov composition conjecture. Vanishing of all the moments $m_k(P,Q)$ and of all the second Melnikov coefficients $D_j(P,Q)$ implies the Composition Condition.

Composition conjecture. The Center and Composition Sets for any polynomial Abel equation coincide.

Let $P$ and $Q$ be polynomials, and consider the polynomial moment equations … A solution is called reducible if there exist polynomials $\widetilde P$, $\widetilde Q$, and $W$ wit…

Let $P(z)=\int p(z)\,dz$ and $Q(z)=\int q(z)\,dz$, where $p$ and $q$ are the polynomial coefficients in the Abel equation … A center is called reducible if there exist polynomials…

Let $P(z)$ and $Q(z)$ be polynomials, and let $a,b$ be points satisfying $P(a)=P(b)$. A solution of system (1) is a polynomial $q(z)=Q'(z)$ satisfying the polynomial moment conditi…

McConnell's conjecture. Every pair of nonconstant polynomials $f_1,f_2\in K[x]$ has a common composite.