Fundamental domain for the Markoff-Hurwitz equation

2023/12/13 by Shin, Eunju
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2312.07890

Abstract

For integers a≠0, k, and n≥3, we consider the Markoff-Hurwitz equation given by x₁¹+⋯+x_n²-ax₁⋯ x_n=k. By defining graphs associated with a height function and by using their properties, we find an exact fundamental domain for a symmetric group generated by involution maps sending (x₁,…,x_n) to (x₁,…,ax₁⋯ x_i-1x_i+1⋯ x_n-x_i,…,x_n), permutations, and double sign changes on the set of integral solutions for the Markoff-Hurwitz equation.

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