On the Gauss-Lucas'lemma in positive characteristic

Umberto Bartocci; Maria Cristina Vipera

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1988)

  • Volume: 82, Issue: 2, page 211-216
  • ISSN: 1120-6330

Abstract

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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 μ ≤ k ≤ n , where μ is the multiplicity of α , then f ( k ) ⁢ ( α ) ≠ 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 > 2 ⁢ n ⁢ — ⁢ 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.

How to cite

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Bartocci, 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 Lincei. Matematica e Applicazioni 82.2 (1988): 211-216. <http://eudml.org/doc/287277>.

@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 Lincei. Matematica e Applicazioni},
keywords = {Roots; Polynomials; Fields of characteristic p; algebraically closed field; free root; polynomials},
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/287277},
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 Lincei. Matematica e Applicazioni
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; algebraically closed field; free root; polynomials
UR - http://eudml.org/doc/287277
ER -

References

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  1. AHLFORS, L.V., Complex Analysis, Mc Graw and Hill, 1966. Zbl0154.31904MR510197
  2. BARTOCCI, U., Su di una congettura di Sudbery, «Rend. Acc. Naz. Lincei», VIII, 56, 1974. Zbl0317.12103MR389865
  3. HARDY, G.H. and WRIGHT, E.M., An Introduction to the Theory of Numbers, Oxford, 1960. Zbl0086.25803MR67125JFM64.0093.03
  4. HAYASHI, T., Relations between the zeros of a rational integral function and its derivate, «Ann. of Math.», 15, 1913. JFM45.0167.01
  5. HILLS, E., Analytic Function Theory, Vol. I, Chelsea P.C., 1973. Zbl0273.30002
  6. IRWIN, F., Relations between the roots of a rational integral function and its derivatives, «Ann. of Math.», 16, 1915. JFM45.0167.02
  7. 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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