BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Chahat Ahuja (Indraprastha Institute of Information Technology\, I ndia) DTSTART:20260718T120000Z DTEND:20260718T122500Z DTSTAMP:20260710T111435Z UID:CANT2026/76 DESCRIPTION:Title: Visibility of lattice points across polynomial curves\nby Chahat Ahu ja (Indraprastha Institute of Information Technology\, India) as part of C ombinatorial and additive number theory seminar (CANT 2026)\n\nLecture hel d in Science Center in the CUNY Graduate Center (4th floor).\n\nAbstract\n The visibility of lattice points from the origin along a polynomial family of curves constitutes a significant generalization of visibility along st raight lines.\nFollowing the classical notion\, where the density of visib le lattice points equals\n$1/\\zeta(2)$\, and its generalization to monomi al curves of the form $y = ax^b$\,\nwhere the density equals $1/(b+1)$\, w e study a family of polynomial curves defined\nby $$ \n y \\\;=\\\; q\\big l(a_n x^n + a_{n-1}x^{n-1} + \\cdots + a_1 x\\bigr)\,\n$$ where $q$ is a p ositive rational number.\n\nWe introduce a new criterion based on a \\emph {polynomial greatest common divisor\ncondition} that provides a lower boun d on the number of visible lattice points in\n$\\mathbb{N}^2$. Conversely\ , we derive conditions under which a given lattice point\nbecomes the next visible point along such a polynomial curve. Using the\nprinciple of incl usion-exclusion\, we obtain an exact double-sum formula for the\nnumber of pairs $(a\, b) \\leq N$ that are visible with respect to this polynomial\ nfamily. \nFinally\, we extend the framework to related problems and pose several open\nquestions concerning gap distributions and quantitative boun ds for non-visible\npoints. This work provides a broader theoretical found ation for lattice point\nvisibility beyond linear and monomial settings.\n LOCATION:https://researchseminars.org/talk/CANT2026/76/ END:VEVENT END:VCALENDAR