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

TL;DR

This paper improves the upper bound constant for the crossing number of polynomial curve systems on surfaces by optimizing a construction, reducing the previous constant by about 30%, and providing a more precise entropy-based analysis.

Contribution

It introduces an optimized construction for bounding crossing numbers, relaxing symmetry constraints and solving an entropy balance problem to achieve a lower constant.

Findings

Reduced the crossing number upper bound constant by approximately 30%.

Derived a new explicit constant C_* ≈ 1.5805 for the bound.

Extended the topological framework with improved bounds.

Abstract

Baader, J\"org, 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 $9/4=2.25$. 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 $\varepsilon>0$, \[ \mathrm{Cr}\bigl(g,\lfloor g^{1+\alpha}\rfloor\bigr) \ \le\ (C_\star+\varepsilon)\,\alpha^2\, g^{1+2\alpha}(\log g)^2 \qquad (g\ \text{sufficiently large}), \] where \[ C_\star\ =\ \inf_{0<x\le 1/2}\ \frac{2x}{H(x)^2} \ \approx\ 1.5805443269, \qquad H(x)=-x\log x-(1-x)\log(1-x). \] This reduces the previous constant by about $30\%$ while staying within the same topological framework.