Patterns and periodicity in a family of resultants
Journal de Théorie des Nombres de Bordeaux, Tome 21 (2009) no. 1, pp. 215-234.

Étant donné un polynôme $f\left(x\right)\in ℤ\left[x\right]$, unitaire de degré $N$, et un entier positif $\ell$, on peut définir un nouveau polynôme ${f}_{\ell }\left(x\right)\in ℤ\left[x\right]$, unitaire de degré $N$, en élevant chaque racine de $f$ à la puissance $\ell$. Nous généralisons un lemme de Dobrowolski pour montrer que, si $m et $p$ est un nombre premier, alors ${p}^{N\left(m+1\right)}$ divise le réesultant de ${f}_{{p}^{m}}$ et ${f}_{{p}^{n}}$. Nous considérons alors la fonction $\left(j,k\right)↦Res\left({f}_{j},{f}_{k}\right)\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}{p}^{m}$. 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.

Given a monic degree $N$ polynomial $f\left(x\right)\in ℤ\left[x\right]$ and a non-negative integer $\ell$, we may form a new monic degree $N$ polynomial ${f}_{\ell }\left(x\right)\in ℤ\left[x\right]$ by raising each root of $f$ to the $\ell$th power. We generalize a lemma of Dobrowolski to show that if $m and $p$ is prime then ${p}^{N\left(m+1\right)}$ divides the resultant of ${f}_{{p}^{m}}$ and ${f}_{{p}^{n}}$. We then consider the function $\left(j,k\right)↦Res\left({f}_{j},{f}_{k}\right)\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}{p}^{m}$. 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.

Publié le : 2009-08-24
DOI : https://doi.org/10.5802/jtnb.667
@article{JTNB_2009__21_1_215_0,
author = {Kevin G. Hare and David McKinnon and Christopher D. Sinclair},
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},
zbl = {pre05620678},
mrnumber = {2537713},
language = {en},
url = {https://jtnb.centre-mersenne.org/item/JTNB_2009__21_1_215_0/}
}
Kevin G. Hare; David McKinnon; Christopher D. Sinclair. 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. https://jtnb.centre-mersenne.org/item/JTNB_2009__21_1_215_0/

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