2023/09/29 by Wang, Tianhao · 1 citation
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2309.16978
openalex publication_date 2023/09/29 · openalex created_date 2023/10/03 · openalex updated_date 2026/07/28
An ordered pair of smooth conics satisfies the Poncelet triangle condition if there is a triangle inscribed in the first conic and circumscribed in the second conic. Over a finite field \mathbbFq with characteristic greater than 3, Chipalkatti showed that the density of pairs of smooth conics satisfying the Poncelet triangle condition is (1)/(q)+O(q-2). We improve this result, showing that the density is exactly (q-1)/(q2-q+1). We consider the problem of determining the density of pairs of conics satisfying the Poncelet n-gon condition for larger n. We prove a corrected version of a conjecture of Chipalkatti, showing that the proportion of pairs of smooth conics satisfying the Poncelet tetragon condition is (1)/(q) + O(q^-3/2). We show that when n is an odd integer coprime to q, the density of pairs of smooth conics satisfying this condition is (d(n)-1)/(q)+O(q-3/2), where d(n) is the number of divisors of n. More generally, we conjecture that the density of pairs of conics satisfying the Poncelet n-gon condition is d'(n)/q in general, where d'(n) is the number of divisors of n not equal to 1 or 2. Our argument involves analyzing the n-torsion points on a certain elliptic curve over the function field K = \mathbbFq(λ).