Étant donné un polynôme , unitaire de degré , et un entier positif , on peut définir un nouveau polynôme , unitaire de degré , en élevant chaque racine de à la puissance . Nous généralisons un lemme de Dobrowolski pour montrer que, si et est un nombre premier, alors divise le réesultant de et . Nous considérons alors la fonction . Nous montrons, pour et fixés, que cette fonction est périodique en et , et exhibons un grand nombre de symétries. Une étude de la structure comme réunion de réseaux est également faite.
Given a monic degree polynomial and a non-negative integer , we may form a new monic degree polynomial by raising each root of to the th power. We generalize a lemma of Dobrowolski to show that if and is prime then divides the resultant of and . We then consider the function . We show that for fixed and that this function is periodic in both and , and exhibits high levels of symmetry. Some discussion of its structure as a union of lattices is also given.
@article{JTNB_2009__21_1_215_0, author = {Hare, Kevin G. and McKinnon, David and Sinclair, Christopher D.}, title = {Patterns and periodicity in a family of resultants}, journal = {Journal de Th\'eorie des Nombres de Bordeaux}, pages = {215--234}, publisher = {Universit\'e Bordeaux 1}, volume = {21}, number = {1}, year = {2009}, doi = {10.5802/jtnb.667}, mrnumber = {2537713}, language = {en}, url = {http://archive.numdam.org/item/JTNB_2009__21_1_215_0/} }
Hare, Kevin G.; McKinnon, David; Sinclair, Christopher D. Patterns and periodicity in a family of resultants. Journal de Théorie des Nombres de Bordeaux, Tome 21 (2009) no. 1, pp. 215-234. doi : 10.5802/jtnb.667. http://archive.numdam.org/item/JTNB_2009__21_1_215_0/
[1] E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial. Acta. Arith. 34 (1979), 391–401. | MR 543210 | Zbl 0416.12001
[2] David S. Dummit, Richard M. Foote, Abstract algebra. Prentice Hall Inc., Englewood Cliffs, NJ, 1991. | MR 1138725 | Zbl 0751.00001
[3] Rudolf Lidl, Harald Niederreiter, Introduction to finite fields and their applications. Cambridge University Press, Cambridge, 1986. | MR 860948 | Zbl 0629.12016
[4] Jürgen Neukirch, Algebraic number theory. Volume 322 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher, with a foreword by G. Harder. | MR 1697859 | Zbl 0956.11021