# Patterns and periodicity in a family of resultants

• [1] Department of Pure Mathematics University of Waterloo Waterloo, Ontario
• [2] Department of Mathematics University of Colorado at Boulder Boulder, Colorado 80309
• Volume: 21, Issue: 1, page 215-234
• ISSN: 1246-7405

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## Abstract

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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 $m<n$ 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.

## How to cite

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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 -

## References

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1. E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial. Acta. Arith. 34 (1979), 391–401. Zbl0416.12001MR543210
2. David S. Dummit, Richard M. Foote, Abstract algebra. Prentice Hall Inc., Englewood Cliffs, NJ, 1991. Zbl0751.00001MR1138725
3. Rudolf Lidl, Harald Niederreiter, Introduction to finite fields and their applications. Cambridge University Press, Cambridge, 1986. Zbl0629.12016MR860948
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. Zbl0956.11021MR1697859

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