Sharp bounds for the number of roots of univariate fewnomials

Abstract

Let K be a field and t ≥ 0. Denote by B m ( 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 B m ( t, L ) ≤ t 2 B m ( t, K ) for any local field L with a non-archimedean valuation v : L → R ∪{∞} such that v | Z 6 =0 ≡ 0 and residue field K , and that B m ( t, K ) ≤ ( t 2 − t +1)( p f − 1) for any finite extension K/ Q p with residual class degree f and ramification index e , assuming that p > t + e . For any finite extension K/ Q p , for p odd, we also show the lower bound B m ( t, K ) ≥ (2 t − 1)( p f − 1), which gives the sharp estimation B m (2 , K ) = 3( p f − 1) for trinomials when p > 2 + e .