On the Hilbert number for piecewise linear vector fields with algebraic discontinuity set

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 $n\geq 4$.