On the Gauss-Lucas'lemma in positive characteristic
Umberto Bartocci; Maria Cristina Vipera
- Volume: 82, Issue: 2, page 211-216
- ISSN: 0392-7881
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topBartocci, Umberto, and Vipera, Maria Cristina. "On the Gauss-Lucas'lemma in positive characteristic." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti 82.2 (1988): 211-216. <http://eudml.org/doc/289144>.
@article{Bartocci1988,
abstract = {If $f(x)$ is a polynomial with coefficients in the field of complex numbers, of positive degree $n$, then $f(x)$ has at least one root a with the following property: if $\mu \le k \le n$, where $\mu$ is the multiplicity of $\alpha$, then $f^\{(k)\} (\alpha) \ne 0$ (such a root is said to be a "free" root of $f(x)$). This is a consequence of the so-called Gauss-Lucas'lemma. One could conjecture that this property remains true for polynomials (of degree $n$) with coefficients in a field of positive characteristic $p > n$ (Sudbery's Conjecture). In this paper it is shown that, on the contrary, if $n > p > 2n—2$ then there exist polynomials which do not have free roots at all. Then one replaces Sudbery's conjecture by supposing that the required property is true for simple polynomials.},
author = {Bartocci, Umberto, Vipera, Maria Cristina},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti},
keywords = {Roots; Polynomials; Fields of characteristic p},
language = {eng},
month = {6},
number = {2},
pages = {211-216},
publisher = {Accademia Nazionale dei Lincei},
title = {On the Gauss-Lucas'lemma in positive characteristic},
url = {http://eudml.org/doc/289144},
volume = {82},
year = {1988},
}
TY - JOUR
AU - Bartocci, Umberto
AU - Vipera, Maria Cristina
TI - On the Gauss-Lucas'lemma in positive characteristic
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
DA - 1988/6//
PB - Accademia Nazionale dei Lincei
VL - 82
IS - 2
SP - 211
EP - 216
AB - If $f(x)$ is a polynomial with coefficients in the field of complex numbers, of positive degree $n$, then $f(x)$ has at least one root a with the following property: if $\mu \le k \le n$, where $\mu$ is the multiplicity of $\alpha$, then $f^{(k)} (\alpha) \ne 0$ (such a root is said to be a "free" root of $f(x)$). This is a consequence of the so-called Gauss-Lucas'lemma. One could conjecture that this property remains true for polynomials (of degree $n$) with coefficients in a field of positive characteristic $p > n$ (Sudbery's Conjecture). In this paper it is shown that, on the contrary, if $n > p > 2n—2$ then there exist polynomials which do not have free roots at all. Then one replaces Sudbery's conjecture by supposing that the required property is true for simple polynomials.
LA - eng
KW - Roots; Polynomials; Fields of characteristic p
UR - http://eudml.org/doc/289144
ER -
References
top- AHLFORS, L.V., Complex Analysis, Mc Graw and Hill, 1966. Zbl0154.31904MR510197
- BARTOCCI, U., Su di una congettura di Sudbery, «Rend. Acc. Naz. Lincei», VIII, 56, 1974. Zbl0317.12103MR389865
- HARDY, G.H. and WRIGHT, E.M., An Introduction to the Theory of Numbers, Oxford, 1960. Zbl0086.25803MR67125JFM64.0093.03
- HAYASHI, T., Relations between the zeros of a rational integral function and its derivate, «Ann. of Math.», 15, 1913. JFM45.0167.01
- HILLS, E., Analytic Function Theory, Vol. I, Chelsea P.C., 1973. Zbl0273.30002
- IRWIN, F., Relations between the roots of a rational integral function and its derivatives, «Ann. of Math.», 16, 1915. JFM45.0167.02
- SUDBERY, A., The number of distinct roots of a polynomial and its derivatives, «Bull. London Math. Soc.», 5, 1973. Zbl0271.12101MR320288
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