Sharp bounds for the number of roots of univariate fewnomials

Details Sharp bounds for the number of roots of univariate fewnomials Sharp bounds for the number of roots of univariate fewnomials

2010-08-27

arxiv.org/abs/1008.4808v2

Mathematics

Number Theory

Scientific paper

Let K be a field and t>=0. Denote by Bm(t,K) the maximum number of non-zero roots in K, counted with multiplicities, of 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)<=t^2 Bm(t,K) for any local field L with a non-archimedean valuation v such that v(n)=0 for all non-zero integer n and residue field K, and that Bm(t,K)<=(t^2-t+1)(p^f-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)(p^f-1), which gives the sharp estimation Bm(2,K)=3(p^f-1) for trinomials when p>2+e.

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