Growth estimate for the number of crossing limit cycles in planar piecewise polynomial vector fields

Motivated by the classical Hilbert's Sixteenth Problem, we extend some main developments obtained for Hilbert's number in the polynomial setting to the piecewise polynomial context. Specifically, we study the growth of the maximum number of crossing limit cycles in planar piecewise polynomial vector fields of degree $n$, denoted by $H_c(n)$. The best previously known general lower bound is $H_c(n)\geq 2n - 1$. In this work, we show that $H_c(n)$ grows at least as fast as $n^2/4.$ Furthermore, we prove that $H_c(n)$ is strictly increasing whenever it is finite, and that in such cases this maximum can be realized by piecewise polynomial systems whose crossing limit cycles are all hyperbolic. Finally, for the more restrictive class of piecewise polynomial Hamiltonian vector fields, we adapt the recursive construction of Christopher and Lloyd to demonstrate that the corresponding maximal number of crossing limit cycles, denoted by $\widehat{H}_c(n)$, grows at least as fast as $n\log n/(2\log 2)$, thereby improving previously established linear growth estimate.