The ``abc'' conjecture for \(p\)-adic functions of several variables (Q1885173)

Let \(a_1(z)\), \(a_2(z)\), \(a_3(z)\) be entire functions on \(\mathbb{C}_p\) and without common zeros and not all constants such that \(a_1(z)+a_2 (z)=a_3(z)\). \textit{P.-C. Hu} and \textit{C.-C. Yang} proved the analogue of ``\(abc\)'' conjecture for one variable non-Archimedean holomorphic functions is true, namely, it holds that \(\max\{T(r,a_1), T(r,a_2), T(r, a_3)\} \leq\overline N(r,1/a_1a_2a_3) -\log r+0(1)\), see [The ``\(abc\)'' conjecture over function fields. Proc. Japan Acad., Ser. A 76, No. 7, 118--120 (2000; Zbl 0979.11023)]. The authors generalized this result to \(p\)-adic entire functions in several variables. Let \(a(z_{(m)})\), \(b(z_{(m)})\), \(c(z_{(m)})\) be entire functions on \(\mathbb{C}_p^m\) and without common zeros and not all constants such that \(a(z_{(m)})+ b(z_{(m)})= c(z_{(m)})\). Suppose that every zero of \(a(z_{(m)})\), \(b(z_{(m)})\), \(c(z_{(m)})\) satisfies the conditions that either all its partial multiplicities are less than or equal to one, or all its partial orders are at least two and not equal to \(+\infty\). Then \[ \max\bigl\{H_\alpha (r_{(m)}), H_b(r_{(m)}), H_b(r_{(m)}) \bigr\} \leq\overline N_{abc} (r_{(m)})-\log R+O(1), \] where \(O(1)\) is bounded when \(R=\min_{1\leq i\leq m}r_i\to +\infty\).