[Paper Review] Distinct values of bilinear forms on algebraic curves

Claudiu Valculescu, Frank de Zeeuw|arXiv (Cornell University)|Mar 16, 2014

Algebraic Geometry and Number Theory11 references3 citations

TL;DR

This paper establishes a lower bound of $\Omega_d(|S|^{4/3})$ on the number of distinct values of a bilinear form $B_M(p,q) = p^T M q$ over finite sets $S$ on irreducible algebraic curves in $\mathbb{C}^2$, provided the curve is not special (i.e., not a line or linearly equivalent to $x^k = y^\ell$ with $\gcd(k,\ell)=1$). The proof adapts and refines techniques from Pach and de Zeeuw's work on Euclidean distances, leveraging automorphism groups and incidence geometry to show that only special curves allow for linearly many values, thus identifying the extremal cases.

ABSTRACT

Let $B$ be a bilinear form on pairs of points in the complex plane, of the form $B(p,q) = p^TMq$, for an invertible $2 imes2$ complex matrix $M$. We prove that any finite set $S$ contained in an irreducible algebraic curve $C$ of degree $d$ in $\mathbb{C}^2$ determines at least $c_d|S|^{4/3}$ distinct values of $B$, unless the curve $C$ has an exceptional form. This strengthens a result of Charalambides in several ways. The proof is based on that of Pach and De Zeeuw, who proved a similar statement for the Euclidean distance function in the real plane. Our main motivation for this paper is that for bilinear forms, this approach becomes more natural, and should better lend itself to understanding and generalization.

Motivation & Objective

• To extend and strengthen Charalambides's result on distinct triangle areas (via bilinear form $B_A$) from $\Omega(|S|^{5/4})$ to $\Omega(|S|^{4/3})$ for real curves, and generalize it to complex curves and arbitrary invertible bilinear forms.
• To identify the complete class of exceptional curves—called 'special curves'—on which bilinear forms can take only $O(|S|)$ distinct values, thus characterizing the extremal cases.
• To improve the dependence on the curve degree $d$ in the lower bound, achieving $\Omega(d^{-14/3}|S|^{4/3})$ for non-special curves, and to clarify the limits of the incidence-theoretic method used in prior work.
• To investigate whether the exponent $4/3$ in the lower bound is tight, and to explore generalizations to higher-degree polynomial functions and higher-dimensional varieties.

Proposed method

• The proof uses an incidence geometry framework based on the setup of Pach and de Zeeuw, adapted for bilinear forms in $\mathbb{C}^2$.
• It relies on analyzing the automorphism group of the curve: if the curve has $O(d^2)$ automorphisms, the bound follows from a bipartite incidence theorem (Theorem 2.1).
• The key step is showing that only special curves (lines or $x^k = y^\ell$ with $\gcd(k,\ell)=1$) have more than $O(d^2)$ automorphisms, using Hurwitz's theorem and degree bounds on rational maps.
• The argument uses dual incidence problems: for each $p \in S$, the set of $q$ such that $B_M(p,q)$ is fixed lies on a line, and the number of such incidences is bounded via a Szemerédi-Trotter-type incidence bound.
• The method explicitly tracks the dependence on the curve degree $d$, improving over prior work by deriving $d^{-14/3}$ as the degree penalty in the bound.
• The framework is shown to be more natural and generalizable for bilinear forms than for the Euclidean distance function, due to the algebraic structure of the forms.

Experimental results

Research questions

• RQ1 What is the minimal number of distinct values a bilinear form $B_M(p,q) = p^T M q$ can take on a finite set $S$ contained in an irreducible algebraic curve $C \subset \mathbb{C}^2$ of degree $d$?
• RQ2 Which curves allow bilinear forms to take only $O(|S|)$ distinct values, and can this class be fully characterized?
• RQ3 Can the exponent $4/3$ in the lower bound $\Omega(|S|^{4/3})$ be improved, or is it tight for this class of functions?
• RQ4 How does the dependence on the curve degree $d$ affect the lower bound, and can this be optimized?
• RQ5 To what extent can the incidence-theoretic method used for Euclidean distances be generalized and simplified for bilinear forms?

Key findings

• For any finite set $S$ on a non-special irreducible algebraic curve $C \subset \mathbb{C}^2$ of degree $d$, the number of distinct values of any invertible bilinear form $B_M$ is at least $\Omega(d^{-14/3}|S|^{4/3})$.
• The only curves on which a bilinear form can take $O(|S|)$ distinct values are the 'special curves'—lines or curves linearly equivalent to $x^k = y^\ell$ with $\gcd(k,\ell)=1$.
• The class of exceptional curves is sharp: for any such curve, there exists a bilinear form $B_M$ such that $|\mathcal{B}_M(S)| = O(|S|)$, showing the bound cannot be improved without excluding these curves.
• The proof method, based on incidence geometry and automorphism group analysis, is more natural and streamlined for bilinear forms than for the Euclidean distance function, and generalizes well to complex curves.
• Over $\mathbb{R}^2$, the same method yields a better degree dependence of $d^{-2}$, leading to a non-vacuous interpolation bound $|F(S)| = \Omega(|S|^{1/3})$.
• The exponent $4/3$ is not expected to be tight; the paper suggests that improvements may come from exploiting the Cartesian product structure of the point set and the restricted family of dual curves in the incidence problem.