2021/05/06 by Pranabesh Das, Shanta Laishram, Das, Pranabesh +5
Mathematics · #Algebraic Geometry and Number Theory #Geometry and complex manifolds #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2105.02792
For the superelliptic curves of the form (x+1) ⋯(x+i-1)(x+i+1)⋯ (x+k)=y^ℓ with x,y ∈ ℚ, y≠ 0, k ≥ 3, 1≤ i≤ k, ℓ ≥ 2, a prime, Das, Laishram, Saradha, and Edis showed that the superelliptic curve has no rational points for ℓ≥ e3k. In fact, the double exponential bound, obtained in these papers is far from reality. In this paper, we study the superelliptic curves for small values of k. In particular, we explicitly solve the above equation for 4 ≤ k ≤ 8.