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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.15877 |
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| _version_ | 1866914275704963072 |
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| author | Ahuja, Chahat |
| author_facet | Ahuja, Chahat |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15877 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Visibility of Lattice Points across Polynomials Ahuja, Chahat Number Theory 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. |
| title | Visibility of Lattice Points across Polynomials |
| topic | Number Theory |
| url | https://arxiv.org/abs/2601.15877 |