4 problems

Let $X=(P,Q)$ be a planar polynomial vector field of degree $n$, let $\pi(X)$ denote its number of limit cycles, and define the Hilbert number by … where $\mathcal{X}_n$ is the fam…

Let $A$ be a family of arcs, and let $\overline{A}$ denote its closure. For each arc, consider the associated bifurcation function and its number of zeros. Uniform zero-bound conje…

Let $C$ be a random real plane curve of degree $d$, and let $h(C)$ denote the energy of its oval arrangement. The expectation with respect to the random-curve model is written as…

Constructive bound conjecture. There exists an explicit (constructive) bound of the form