  The second part of the Hilbert's sixteenth problem consists in determining
the upper bound $\mathcal{H}(n)$ for the number of limit cycles that planar
polynomial vector fields of degree $n$ can have. For $n\geq2$, it is still
unknown whether $\mathcal{H}(n)$ is finite or not. The main achievements
obtained so far establish lower bounds for $\mathcal{H}(n)$. Regarding
asymptotic behavior, the best result says that $\mathcal{H}(n)$ grows as fast
as $n^2\log(n)$. Better lower bounds for small values of $n$ are known in the
research literature. In the recent paper "Some open problems in low dimensional
dynamical systems" by A. Gasull, Problem 18 proposes another Hilbert's
sixteenth type problem, namely improving the lower bounds for $\mathcal{L}(n)$,
$n\in\mathbb{N}$, which is defined as the maximum number of limit cycles that
planar piecewise linear differential systems with two zones separated by a
branch of an algebraic curve of degree $n$ can have. So far,
$\mathcal{L}(n)\geq [n/2],$ $n\in\mathbb{N}$, is the best known general lower
bound. Again, better lower bounds for small values of $n$ are known in the
research literature. Here, by using a recently developed second order Melnikov
method for nonsmooth systems with nonlinear discontinuity manifold, it is shown
that $\mathcal{L}(n)$ grows as fast as $n^2.$ This will be achieved by
providing lower bounds for $\mathcal{L}(n)$, which improves every previous
estimates for $n\geq 4$.
