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~\ref{conj}, $d\ge 3,$ and $q$ and $p$ are prime numbers, where $p\le q^A$ for some $A>0$. We develop a polynomial-time algorithm in $\log (q)$ that lifts every $\mathbb{Z}/q\mathbb{Z}$ point of $S^{d-2}\subset S^d$ to a $\mathbb{Z}[1/p]$ point of $S^d$ with the minimum height. We implement our algorithm for $d=3\text{ and }4$. 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 $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$.

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