Zeros of sparse polynomials over local fields of characteristic~$p$
Bjorn Poonen · Mathematical Research Letters · 1998
Let K be a field of characteristic p> 0 equipped with a valuation v: K ∗ → G taking values in an ordered abelian group G. Let OK = {α ∈ K: v(α) ≥ 0} and mK = {α ∈ K: v(α)> 0} be the valuation ring and maximal ideal, respectively, and suppose that the residue field OK/mK is finite, with q elements. Theorem 1. If f(x) = a0xn0 + a1xn1 + · · · + akxnk is a polynomial with k + 1 nonzero coefficients ai ∈ K ∗ , then f has at most qk distinct zeros in K. This upper bound is sharp: if K is Fq((T)) with the usual discrete valuation v: K ∗ → Z, if V ⊂ K is an Fq-subspace of dimension k, and if c ∈ K is nonzero, then the polynomial f(x): = c ∏ α∈V (x − α) has the form a0x + a1xq + · · · + akxqk for some a0, a1,..., ak ∈ K ∗. Theorem 1 is the case d = 1 of the following generalization, which bounds the number of distinct zeros of bounded degree. Let µ(n) be the Möbius µ-function. Theorem 2. Fix d ≥ 1. If f(x) = a0xn0 + a1xn1 + · · · + akxnk is a polynomial with k + 1 nonzero coefficients ai ∈ K ∗ , then the number of distinct zeros of f in K of degree at most d over K is at most ∑d