Sur les zéros des fonctions symétriques complètes des corps cubiques. (On the zeros of complete symmetric functions in cubic fields)
Let \({\mathcal K}\) be a real cubic field of negative discriminant, defined by the relation \(\omega\) \(3=\epsilon Norm_{{\mathcal K}/{\mathbb{Q}}}(\omega)\), where \(\omega\) is an integer of \({\mathcal K}\) and the unit \(\epsilon\) is not a cube. The complete symmetric function with \(weight\quad n\) of the roots of the minimal polynomial of \(\omega\) satisfies a cubic linear recurrence relation. We show that the zeros of this sequence are connected to the binomial units of \({\mathcal K}\) and further, that the number of zeros is between two and four, except in two cases, when there are five or six zeros (the sequence of Berstel and Mignotte). As an illustration, a table of zeros is obtained for fields \({\mathcal K}\) of discriminant superior to - 400.
symmetric function of the roots of minimal polynomial
real cubic field of negative discriminant
scientific article; zbMATH DE number 4045876