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: Avendaño, M., Krick, T.
Formato: JOUR
Materias:
Generalized vandermonde determinants
Lacunary polynomials
Local fields
Root counting
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_0022314X_v131_n7_p1209_Avendano
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling todo:paper_0022314X_v131_n7_p1209_Avendano2023-10-03T14:31:05Z Sharp bounds for the number of roots of univariate fewnomials Avendaño, M. Krick, T. Generalized vandermonde determinants Lacunary polynomials Local fields Root counting 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. Fil:Avendaño, M. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Krick, T. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_0022314X_v131_n7_p1209_Avendano
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Generalized vandermonde determinants
Lacunary polynomials
Local fields
Root counting
spellingShingle Generalized vandermonde determinants
Lacunary polynomials
Local fields
Root counting
Avendaño, M.
Krick, T.
Sharp bounds for the number of roots of univariate fewnomials
topic_facet Generalized vandermonde determinants
Lacunary polynomials
Local fields
Root counting
description 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.
format JOUR
author Avendaño, M.
Krick, T.
author_facet Avendaño, M.
Krick, T.
author_sort Avendaño, M.
title Sharp bounds for the number of roots of univariate fewnomials
title_short Sharp bounds for the number of roots of univariate fewnomials
title_full Sharp bounds for the number of roots of univariate fewnomials
title_fullStr Sharp bounds for the number of roots of univariate fewnomials
title_full_unstemmed Sharp bounds for the number of roots of univariate fewnomials
title_sort sharp bounds for the number of roots of univariate fewnomials
url http://hdl.handle.net/20.500.12110/paper_0022314X_v131_n7_p1209_Avendano
work_keys_str_mv AT avendanom sharpboundsforthenumberofrootsofunivariatefewnomials
AT krickt sharpboundsforthenumberofrootsofunivariatefewnomials
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