Unlikely intersections for curves in multiplicative groups over positive characteristic (Q2922435)

This paper investigates the phenomenon of the intersection inside a commutative group variety of an algebraic subvariety and a union of group subvarieties. Such intersections have been widely investigated in characteristic zero (where major conjectures are the Manin-Mumford conjecture and the Mordell-Lang conjecture), and this paper is a contribution to the study of such intersections in characteristic \(p\).NEWLINENEWLINEIn particular, the author makes the following conjecture: Suppose \(K\) is an algebraically closed field of characteristic \(p > 0\), and let \(C\) in \(\mathbb{G}_m^n\) be an irreducible curve defined over \(K\). Assume that for any linearly independent \((r_1, \ldots, r_n)\), \((s_1, \ldots, s_n)\) in \(\mathbb{Z}^n\), the monomials \(x_1^{r_1} \cdots x_n^{r_n}\) and \(x_1^{s_1} \cdots x_n^{s_n}\) are algebraically independent over \(\mathbb{F}_p\) on \(C\). Then there are at most finitely many \((\xi_1, \ldots, \xi_n)\) in \(C(K)\) for which there exist linearly independent \((a_1, \ldots, a_n)\), \((b_1, \ldots, b_n)\) in \(\mathbb{Z}^n\) such that \(\xi_1^{a_1} \cdots \xi_n^{a_n} = \xi_1^{b_1} \cdots \xi_n^{b_n} = 1\).NEWLINENEWLINEThe above conjecture is true in characteristic zero under the weaker hypothesis that \(x_1^{r_1} \cdots x_n^{r_n}\) is never identically \(1\) on \(C\) for \((r_1, \ldots, r_n)\) non-zero. However, it fails in characteristic \(p\) without a stronger hypothesis (actually, the author gives a weaker hypothesis which may suffice, but which is still stronger than the hypothesis in characteristic zero). The conjecture is proven for \(n = 3\), and for arbitrary \(n\) when \(C\) is of a specific type (it is trivial when \(n = 2\)). The proofs in both cases use the technique of logarithmically differentiating the equation that needs to hold with respect to a transcendental variable.NEWLINENEWLINEOne typo worth pointing out: in the fifth line of Theorem 1.1, it should say ``are algebraically \textit{dependent} over \(\mathbb{F}_p\) on \(C\).''