Artículo
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{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.
Registro:
| Documento: | Artículo |
| Título: | Sharp bounds for the number of roots of univariate fewnomials |
| Autor: | Avendaño, M.; Krick, T. |
| Filiación: | Texas A and M University, Department of Mathematics, Milner Bldg. 023, College Station, TX 77843-3368, United States Departamento de Matemática, FCEyN, Universidad de Buenos Aires and CONICET, Ciudad Universitaria, 1428, Buenos Aires, Argentina |
| Palabras clave: | Generalized vandermonde determinants; Lacunary polynomials; Local fields; Root counting |
| Año: | 2011 |
| Volumen: | 131 |
| Número: | 7 |
| Página de inicio: | 1209 |
| Página de fin: | 1228 |
| DOI: | http://dx.doi.org/10.1016/j.jnt.2011.01.006 |
| Título revista: | Journal of Number Theory |
| Título revista abreviado: | J. Number Theory |
| ISSN: | 0022314X |
| CODEN: | JNUTA |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022314X_v131_n7_p1209_Avendano |
Referencias:
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- Avendaño, M., Krick, T., Pacetti, A., Newton-Hensel interpolation lifting (2006) Found. Comput. Math., 6 (1), pp. 81-120
- Bochnak, J., Coste, M., Roy, M.-F., (1998) Real Algebraic Geometry, , Springer-Verlag
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- Weiss, E., (1963) Algebraic Number Theory, , McGraw-Hill
Citas:
---------- APA ----------
Avendaño, M. & Krick, T. (2011) . Sharp bounds for the number of roots of univariate fewnomials. Journal of Number Theory, 131(7), 1209-1228.http://dx.doi.org/10.1016/j.jnt.2011.01.006
---------- CHICAGO ----------
Avendaño, M., Krick, T. "Sharp bounds for the number of roots of univariate fewnomials" . Journal of Number Theory 131, no. 7 (2011) : 1209-1228.http://dx.doi.org/10.1016/j.jnt.2011.01.006
---------- MLA ----------
Avendaño, M., Krick, T. "Sharp bounds for the number of roots of univariate fewnomials" . Journal of Number Theory, vol. 131, no. 7, 2011, pp. 1209-1228.http://dx.doi.org/10.1016/j.jnt.2011.01.006
---------- VANCOUVER ----------
Avendaño, M., Krick, T. Sharp bounds for the number of roots of univariate fewnomials. J. Number Theory. 2011;131(7):1209-1228.http://dx.doi.org/10.1016/j.jnt.2011.01.006
