Patterns and periodicity in a family of resultants

Hare, Kevin G.; McKinnon, David; Sinclair, Christopher D.

Hare, Kevin G.; McKinnon, David; Sinclair, Christopher D.

Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, p. 215-234

Abstract
Résumé

Given a monic degree $N$ polynomial $f (x) \in ℤ [x]$ and a non-negative integer $ℓ$ , we may form a new monic degree $N$ polynomial $f_{ℓ} (x) \in ℤ [x]$ by raising each root of $f$ to the $ℓ$ 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 $(j, k) \mapsto Res (f_{j}, f_{k}) mod p^{m}$ . 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 $f (x) \in ℤ [x]$ , unitaire de degré $N$ , et un entier positif $ℓ$ , on peut définir un nouveau polynôme $f_{ℓ} (x) \in ℤ [x]$ , unitaire de degré $N$ , en élevant chaque racine de $f$ à la puissance $ℓ$ . 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 $(j, k) \mapsto Res (f_{j}, f_{k}) mod p^{m}$ . 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.

MR 2537713 | Zbl pre05620678

DOI : https://doi.org/10.5802/jtnb.667

@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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