Geodesic
On the Gauss-Lucas'lemma in positive characteristic
Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 82 (1988) no. 2, pp. 211-216 Cet article a éte moissonné depuis la source Biblioteca Digitale Italiana di Matematica

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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 $\mu \le k \le n$, 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. 
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     title = {On the {Gauss-Lucas'lemma} in positive characteristic},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali},
     pages = {211--216},
     year = {1988},
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Bartocci, Umberto; Vipera, Maria Cristina. On the Gauss-Lucas'lemma in positive characteristic. Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 82 (1988) no. 2, pp. 211-216. http://geodesic.mathdoc.fr/item/RLINA_1988_8_82_2_a1/

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