13 problems

Two-configuration conjecture. There are only two types of configurations of centers for such a vector field.

A Lie algebra of vector fields with polynomial coefficients is a Lie algebra whose elements are polynomial-coefficient vector fields, with the usual commutator bracket. Leites' fin…

Let $\dot{x}=f(x)$ be a globally asymptotically stable homogeneous polynomial vector field, and let a homogeneous polynomial Lyapunov function mean a homogeneous polynomial certify…

Consider the polynomial system denoted by (gs), its Bautin ideal $B$, and the associated parameter-space vector field denoted by (Vvfsh). Bautin-variety invariance conjecture. The…

Let ${\bf W}$ be a component of the integrability variety ${\bf V}(B)$ of the system under consideration. Let $\mathcal A|_{\bf W}$ denote the restriction to ${\bf W}$ of the opera…

Consider the $1:-1$ resonant polynomial system … Let $I_S$ be its Sibirsky ideal and let $B$ be the Bautin ideal, with ${\bf V}(I_S)$ and ${\bf V}(B)$ their varieties. Sibirsky ide…

Consider the quartic vector field on the Poincaré sphere studied in the paper, and suppose that an annulus formed by a continuum of periodic solutions crosses the equator. Global-c…

Let $P_n(x,y)$ and $Q_n(x,y)$ be polynomials of degree $n$, defining the planar polynomial system … An invariant line is a straight line preserved by the flow of this system. Ye's…

Arnold's undecidability conjecture. There is no algorithm for deciding stability of polynomial vector fields with rational coefficients in some sufficiently high degree and dimensi…

Conjecture. Under assumptions (H1) and (Hk), for some $k\geq 2$, the polynomial field has no elementary first integral.

Let $d\in\mathbb{N}^*$ with $d\geq 2$, and let $X$ be a polynomial vector field of degree $d$. An isochronous condition has geometric complexity $(q,m)$ if it admits a representati…

Consider a real planar polynomial vector field of degree $m$, equivalently a system … where $p,q\in\mathbb R[x,y]$ and $\max\{\deg p,\deg q\}=m$. An algebraic limit cycle is a limi…

Artés's conjecture. The maximum number satisfies