Patterns and periodicity in a family of resultants

Hare, Kevin G., McKinnon, David, and Sinclair, Christopher D.. "Patterns and periodicity in a family of resultants." Journal de Théorie des Nombres de Bordeaux 21.1 (2009): 215-234. <http://eudml.org/doc/10873>.

@article{Hare2009,
abstract = {Given a monic degree $N$ polynomial $f(x) \in \mathbb\{Z\}[x]$ and a non-negative integer $\ell $, we may form a new monic degree $N$ polynomial $f_\{\ell \}(x) \in \mathbb\{Z\}[x]$ by raising each root of $f$ to the $\ell $th power. We generalize a lemma of Dobrowolski to show that if $m &lt; n$ 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 $(j,k) \mapsto \operatorname\{Res\}(f_j, f_k) \;\@mod \;p^m$. 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.},
affiliation = {Department of Pure Mathematics University of Waterloo Waterloo, Ontario; Department of Pure Mathematics University of Waterloo Waterloo, Ontario; Department of Mathematics University of Colorado at Boulder Boulder, Colorado 80309},
author = {Hare, Kevin G., McKinnon, David, Sinclair, Christopher D.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {lemma of Dobrowolski; resultant; periodicity},
language = {eng},
number = {1},
pages = {215-234},
publisher = {Université Bordeaux 1},
title = {Patterns and periodicity in a family of resultants},
url = {http://eudml.org/doc/10873},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Hare, Kevin G.
AU - McKinnon, David
AU - Sinclair, Christopher D.
TI - Patterns and periodicity in a family of resultants
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 1
SP - 215
EP - 234
AB - Given a monic degree $N$ polynomial $f(x) \in \mathbb{Z}[x]$ and a non-negative integer $\ell $, we may form a new monic degree $N$ polynomial $f_{\ell }(x) \in \mathbb{Z}[x]$ by raising each root of $f$ to the $\ell $th power. We generalize a lemma of Dobrowolski to show that if $m &lt; n$ 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 $(j,k) \mapsto \operatorname{Res}(f_j, f_k) \;\@mod \;p^m$. 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.
LA - eng
KW - lemma of Dobrowolski; resultant; periodicity
UR - http://eudml.org/doc/10873
ER -