Diophantine inequalities for the non-Archimedean line \({\mathbb F}_q ((1/T))\) (Q2717600)

The author here continues his studies [see J. Number Theory 78, 46-61 (1999; Zbl 1029.11050)] of a generalization of the Hardy-Littlewood method applied to the completion \(K_{\infty}=\mathbb{F}_q((1/T))\) of the rational function field \(K=\mathbb{F}_{q}(T)\) at the infinite place, where \(\mathbb{F}_q\) denotes the finite field with \(q=p^n\) elements (\(p\) a prime). The main result of the paper is: Suppose that \(d,D,m\) are positive integers and \(\lambda,\lambda_1 ,\cdots , \lambda_D \) are non-zero elements of \(K_{\infty}\) satisfying \(\lambda_1/\lambda_2 \not\in K\), \(2\leq d\leq p\), \(\text{deg}\lambda_1 =\dots = \text{deg}\lambda_D =0\) and \(\text{sgn }\lambda_1+\cdots + \text{sgn }\lambda_D=0\). If NEWLINE\[NEWLINED\geq 1+2^d \text{ for } 2\leq d<11, \text{ or } D\geq 2[2d^2\text{ln }d + d^2\text{ln ln } d + 2d^2 - 2d] + 1\text{ for } d\geq 11,NEWLINE\]NEWLINE then there exist infinitely many positive integers \(N\) for which there are NEWLINE\[NEWLINE\gg {q^{(D-d)}N}/{N^{D+m}} NEWLINE\]NEWLINE \(D\)-tuples \((P_1,\cdots, P_D)\) of monic polynomials with \(\text{deg}(\lambda_iP_i)=N\) and NEWLINE\[NEWLINE\text{deg}(\lambda + \lambda_1P_{1}^{d} +\cdots + \lambda_{D}P_{D}^{d}) < -m \text{ln }N + 1,NEWLINE\]NEWLINE where the implied constant depends only on \(\mathbb{F}_{q}[T], \lambda, \lambda_i, d, D , m\), but not on \(N\).