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Visibility of Lattice Points across Polynomials

Chahat Ahuja

TL;DR

This work generalizes lattice point visibility from straight lines to polynomial curves with nonnegative coefficients by introducing a gcd-based visibility criterion and a fixed polynomial family framework. It proves that any target lattice point can be made visible on a suitable polynomial curve, while also deriving density bounds and exact counting expressions for visible pairs under polynomial-curve constraints. The paper further develops a computational framework to explore blocks of invisible points, reporting initial findings of small invisible blocks within a bounded grid and outlining practical limitations and future computational directions. Collectively, the contributions deepen the theoretical understanding of visibility in the plane beyond linear and monomial settings and link arithmetic properties to geometric patterns on the lattice.

Abstract

The visibility of lattice points from the origin along a polynomial family of curves constitutes a significant generalization of visibility along straight lines. Following the classical notion, where the density equals 1/2, and its generalization to monomial curves of the form y = a x^b, where the density equals 1/(b+1), we study a family of polynomial curves defined by y = q(a_n x^n + ... + a_1 x), where q is a positive rational number. We introduce a new criterion based on a polynomial greatest common divisor condition that provides a lower bound on the number of visible lattice points in N^2. Conversely, we derive conditions under which a given lattice point becomes the next visible point along such a polynomial curve. Using the principle of inclusion-exclusion, we also obtain an exact double-sum formula for the number of pairs (a, b) less than or equal to N that are visible with respect to this polynomial family. Finally, we extend the framework to related problems and pose several open questions concerning gap distributions and quantitative bounds for non-visible points. This work provides a broader theoretical foundation for lattice point visibility beyond linear and monomial settings.

Visibility of Lattice Points across Polynomials

TL;DR

This work generalizes lattice point visibility from straight lines to polynomial curves with nonnegative coefficients by introducing a gcd-based visibility criterion and a fixed polynomial family framework. It proves that any target lattice point can be made visible on a suitable polynomial curve, while also deriving density bounds and exact counting expressions for visible pairs under polynomial-curve constraints. The paper further develops a computational framework to explore blocks of invisible points, reporting initial findings of small invisible blocks within a bounded grid and outlining practical limitations and future computational directions. Collectively, the contributions deepen the theoretical understanding of visibility in the plane beyond linear and monomial settings and link arithmetic properties to geometric patterns on the lattice.

Abstract

The visibility of lattice points from the origin along a polynomial family of curves constitutes a significant generalization of visibility along straight lines. Following the classical notion, where the density equals 1/2, and its generalization to monomial curves of the form y = a x^b, where the density equals 1/(b+1), we study a family of polynomial curves defined by y = q(a_n x^n + ... + a_1 x), where q is a positive rational number. We introduce a new criterion based on a polynomial greatest common divisor condition that provides a lower bound on the number of visible lattice points in N^2. Conversely, we derive conditions under which a given lattice point becomes the next visible point along such a polynomial curve. Using the principle of inclusion-exclusion, we also obtain an exact double-sum formula for the number of pairs (a, b) less than or equal to N that are visible with respect to this polynomial family. Finally, we extend the framework to related problems and pose several open questions concerning gap distributions and quantitative bounds for non-visible points. This work provides a broader theoretical foundation for lattice point visibility beyond linear and monomial settings.
Paper Structure (20 sections, 10 theorems, 71 equations, 1 table, 2 algorithms)

This paper contains 20 sections, 10 theorems, 71 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction and Background
  2. Point to polynomial Visibility
  3. Polynomial to Point Visibility
  4. Open problem using Computational Exploration: Blocks of Invisible Points
  5. Algorithmic Framework
  6. Algorithm Explanation
  7. Polynomial Function
  8. Visibility Evaluation
  9. Breadth-First Search Module
  10. Search within a Defined Region
  11. Complexity Analysis
  12. Time Complexity
  13. Space Complexity
  14. Experimental Setup and Results
  15. System Configuration
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let $(a,b) \in \mathbb{N}^{2}$, and let $\ell$ be a prime number satisfying $\ell > \max(a,b)$. Suppose that the base representation $a$ of $\ell$ is given by where $a_{i}$ are the digits in base-$a$ expansion of $\ell$, i.e., define the polynomial Then the point $(a,b)$ lies on the curve $\mathcal{P}(x)$, and there is no lattice point on the segment of $\mathcal{P}(x)$ that joins $(0,0)$ to $(

Theorems & Definitions (17)

  • Theorem 2.1
  • proof
  • Lemma 2.2
  • Definition 3.1: Polynomial gcd
  • Lemma 3.2
  • proof
  • Theorem 3.3: Chaubey--Pandey--Regavim Sneha
  • Theorem 3.4: Chaubey--Pandey--Regavim Sneha
  • Definition 3.5
  • Theorem 3.6
  • ...and 7 more