Rational and lacunary algebraic curves

TL;DR

The paper bounds the number of intersection points between a fixed rational plane curve and a lacunary algebraic curve $Q=0$ inside a disk, uniformly in $Q$'s coefficients. It builds a Bautin-matrix framework for the parametric family $f_\lambda(z)=Q_\lambda(P(z))$ and performs a blockwise triangulation of the Bautin matrix using lacunarity to avoid block overlap, yielding explicit control of the Bautin index $b$ and the minor bound $\delta$. This leads to a bound $\mathcal{Z}=O(dm)$, with constants depending on the lacunarity diagram and initial jets $\alpha_0,a_0$, and extends to rational curves by reducing to the polynomial case via clearing denominators. The results thus provide uniform, coefficient-insensitive intersection bounds that scale linearly with the degree and the monomial count of the lacunary curve, highlighting lacunarity as a key structural condition.

Abstract

We give a bound on the number $\mathcal{Z}$ of intersection points in a ball of the complex plane, between a rational curve and a lacunary algebraic curve $Q=0$. This bound depends only on the lacunarity diagram of $Q$, and in particular is uniform in the coefficients of $Q$. Our bound shows that $\mathcal{Z}=O(dm)$, where $d$ is the degree of $Q$ and $m$ is the number of its monomials.

Theorems & Definitions (53)

• Remark 1.1
• Definition 1.2
• Proposition 1.4
• proof
• Remark 2.1
• Remark 2.2
• Remark 2.4
• Lemma 2.5
• proof
• Remark 2.6