Zeros of sparse polynomials over local fields of characteristic \(p\) (Q1271366)

Let \(K\) be a field of characteristic \(p>0\) equipped with a valuation \(v:K^*\rightarrow G\) taking values in an ordered abelian group \(G\). Let \({\mathcal O}_K\) and \({\mathfrak m}_K\) be the associated valuation ring and its maximal ideal, respectively, and suppose that the residue field \({\mathcal O}_k/{\mathfrak m}_K\) is finite with \(q\) elements. The author proves that, if \(f(x)=a_0x^{n_0}+a_1x^{n_1}+\dots +a_kx^{n_k}\) is a polynomial with \(k+1\) non-zero coefficients \(a_i\in K^*\), then \(f\) has at most \(q^k\) distinct roots in \(K\). More generally, the number of distinct roots of \(f\) in \(\overline{K}\) of degree at most \(d\) over \(K\) does not exceed \(\sum_{j=1}^d\sum_{i|j} q^{ik}\mu(j/i)\), where \(\mu\) is the Möbius functions. Both bounds are sharp, since equality can be attained, as shown by a number of examples in the case when \(K={\mathbb F}_q((T))\).