Patterns and periodicity in a family of resultants
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, p. 215-234

Given a monic degree N polynomial f(x)[x] and a non-negative integer , we may form a new monic degree N polynomial f (x)[x] by raising each root of f to the th power. We generalize a lemma of Dobrowolski to show that if m<n and p is prime then p N(m+1) divides the resultant of f p m and f p n . We then consider the function (j,k)Res(f j ,f k )modp m . We show that for fixed p and m that this function is periodic in both j and k, and exhibits high levels of symmetry. Some discussion of its structure as a union of lattices is also given.

Étant donné un polynôme f(x)[x], unitaire de degré N, et un entier positif , on peut définir un nouveau polynôme f (x)[x], unitaire de degré N, en élevant chaque racine de f à la puissance . Nous généralisons un lemme de Dobrowolski pour montrer que, si m<n et p est un nombre premier, alors p N(m+1) divise le réesultant de f p m et f p n . Nous considérons alors la fonction (j,k)Res(f j ,f k )modp m . Nous montrons, pour p et m fixés, que cette fonction est périodique en j et k, et exhibons un grand nombre de symétries. Une étude de la structure comme réunion de réseaux est également faite.

@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},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {1},
     year = {2009},
     pages = {215-234},
     doi = {10.5802/jtnb.667},
     mrnumber = {2537713},
     zbl = {pre05620678},
     language = {en},
     url = {http://www.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, Volume 21 (2009) no. 1, pp. 215-234. doi : 10.5802/jtnb.667. http://www.numdam.org/item/JTNB_2009__21_1_215_0/

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