Optimizing the Upper-Bound Constant for the Crossing Number of Polynomial Curve Systems

Baader, Jörg, and Parlier recently established an upper bound for the crossing number of curve systems of size $m\asymp g^{1+\alpha }$ on a genus $g$ surface, obtaining a leading coefficient of $9\mathrm{/}4=2.25$. Their construction relies on fibre surfaces associated with complete bipartite graphs and uses a symmetric parameter choice corresponding to the central binomial coefficient. In this note, we optimize their construction by relaxing the parameter symmetry and solving the resulting entropy balance problem. We show that for every $\alpha >0$ and every $\epsilon >0$, $$\mathrm{C}\mathrm{r}(g,\lfloor g^{1+\alpha }\rfloor )\le (C_{\star }+\epsilon ){\alpha }^2{g}^{1+2\alpha }(\log g{)}^2(g\text{ sufficiently large}),$$ where $$C_{\star }=\underset{0<x\le 1\mathrm{/}2}{\inf }\frac{2x}{H(x{)}^2}\approx 1.5805443269,H(x)=-x\log x-(1-x)\log (1-x)\mathrm{.}$$ This reduces the previous constant by about $30\mathrm{\%}$ while staying within the same topological framework.