12 problems

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…

Homogeneous cubic focal-value conjecture. If no points of $X_5-X_6$ are found in the stated random search, then

Four-component conjecture. There are exactly $4$ codimension-$7$ components of $X_{\infty}$, one of which is $\overline{X_{CR11}}$.

Component-count conjecture. The variety $X_{\infty}$ has exactly $1$ component of codimension $5$, exactly $2$ components of codimension $6$, exactly $4$ components of codimension…

Zoladek's conjecture. All cubic systems with stable solutions near the origin are either of Darboux type or rationally reversible.

Alwash–Lloyd composition-center conjecture. All centers of the Abel equation are composition centers. This conjecture concerns the characterization of periodic behavior in Abel equ…

Let $f$ be a polynomial and consider the Abel equation … A center at $x=0$ means that solutions sufficiently close to $0$ satisfy $x(-1)=x(1)$. No-parameter center conjecture. If,…

Composition condition conjecture. The sufficient condition given in Theorem 1 is also necessary for a polynomial Abel differential equation to have a center on $[a,b]$.

Consider the higher-order Abel equation … with the composition condition that there is a differentiable function $q$ satisfying $q(\omega)=q(0)$ for some $\omega>0$ and … where the…

Composition conjecture. All centers of Abel differential equations with polynomial coefficients are universal; equivalently, for every center of such an equation, the corresponding…

Component-count conjecture. The number of reduced components of the center variety in degree $3$ is $1$ in codimension $5$, $2$ in codimension $6$, $4$ in codimension $7$, $4$ in c…

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…