On the Gauss-Lucas'lemma in positive characteristic

Bartocci, Umberto ; Vipera, Maria Cristina

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

Résumé

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.