The diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$

Abstract

Assume a polynomial-time algorithm for factoring integers, Conjecture 1.1, d ≥ 3, and q and p are prime numbers, where p ≤ q A for some A > 0. We develop a polynomial-time algorithm in log(q) that lifts every Z/qZ point of S d−2 ⊂ S d to a Z[1/p] point of S d with the minimum height. We implement our algorithm for d = 3 and 4. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the Z/qZ points of S d−2 ⊂ S d .