Two Combinatorial Geometric Problems Involving Modular Hyperbolas

TL;DR

This paper establishes a lower bound on the number of ordinary lines formed by a modular hyperbola with prime power modulus and addresses a related open question in the field.

Contribution

It provides new bounds for geometric configurations involving modular hyperbolas and advances understanding of their combinatorial properties.

Findings

Lower bound for ordinary lines on modular hyperbolas

Partial answer to Shparlinski's question

Insights into geometric structures over finite fields

Abstract

We give a lower bound for the number of ordinary lines spanned by a modular hyperbola when the modulus is a prime power. We also give a partial answer to a question of Shparlinski.