Sharp bounds for the number of roots of univariate fewnomials
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...
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| Autores principales: | , |
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| Formato: | JOUR |
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_0022314X_v131_n7_p1209_Avendano |
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| Sumario: | 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{double-struck}∪{∞} such that v|Z{double-struck}≠0≡0 and residue field K, and that Bm(t,K)≤(t2-t+1)(pf-1) for any finite extension K/Q{double-struck}p with residual class degree f and ramification index e, assuming that p>t+e. For any finite extension K/Q{double-struck}p, 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. © 2011 Elsevier Inc. |
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