Sharp bounds for the number of roots of univariate fewnomials |
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Authors: | Martí n Avendañ o,Teresa Krick |
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Affiliation: | a Texas A&M University, Department of Mathematics, Milner Bldg. 023, College Station, TX 77843-3368, USA b Departamento de Matemática, FCEyN, Universidad de Buenos Aires and CONICET, Ciudad Universitaria, 1428, Buenos Aires, Argentina |
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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)?t2Bm(t,K) for any local field L with a non-archimedean valuation v:L→R∪{∞} such that vZ≠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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Keywords: | 11S05 13F30 |
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