Sharp bounds for the number of roots of univariate fewnomials

Abstract

Let K be a field and t ≥ 0. Denote by Bm(t,K) the supremum of the number of roots in K∗, counted with multiplicities, that can have a non-zero polynomial in K[x] with at most t + 1 monomial terms. We prove, using an unified approach based on Vandermonde determinants, that Bm(t, L) ≤ tBm(t,K) for any local field L with a non-archimedean valuation v : L → R ∪ {∞} such that v|Z6=0 ≡ 0 and residue field K, and that Bm(t,K) ≤ (t2− t+1)(pf −1) for any finite extension K/Qp with residual class degree f and ramification index e, assuming that p > t + e. For any finite extension K/Qp, for p odd, we also show the lower bound Bm(t,K) ≥ (2t−1)(pf −1), which gives the sharp estimation Bm(2,K) = 3(pf − 1) for trinomials when p > 2 + e.

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