Poncelet Triangles and Tetragons over Finite Fields

2025/11/09 by Radnović, Milena, Ragas, Ruzzel
Mathematics · #11G20 #11T06 #51E15 #51N15 #51N35 #60C05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory

paper · doi:10.48550/arxiv.2511.06347

openalex publication_date 2025/11/09 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28

Abstract

In the projective plane over a finite field of characteristic not equal to 2, we compute the probability that a randomly selected pair of distinct conics (\mathscrA,\mathscrB), with \mathscrA smooth or singular and \mathscrB smooth, in a fixed pencil of conics will admit a triangle or a tetragon inscribed in \mathscrA and circumscribed about \mathscrB. We do this for all pencils, classified up to projective automorphism, with at least one smooth conic; effectively allowing the case where our conic pairs intersect non-transversally.

Citations

• Poncelet porism in singular cases
• Linear complete symmetric rank-distance codes
• Counting pairs of conics over finite fields that satisfy the Poncelet n-gon condition
• Poncelet's theorem for conics in any position and any characteristic
• On the Poncelet triangle condition over finite fields
• A Poncelet Criterion for special pairs of conics in PG(2,p)
• Poncelet's Theorem in the four nonisomorphic finite projective planes of order 9

Related

• On the Poncelet triangle condition over finite fields
• A Poncelet Criterion for special pairs of conics in PG(2,p)
• Counting pairs of conics over finite fields that satisfy the Poncelet n-gon condition
• Poncelet's theorem for conics in any position and any characteristic
• Conics associated with triangles, or how Poncelet meets Morley