# Patterns and periodicity in a family of resultants

Kevin G. Hare^{[1]}; David McKinnon^{[1]}; Christopher D. Sinclair^{[2]}

- [1] Department of Pure Mathematics University of Waterloo Waterloo, Ontario
- [2] Department of Mathematics University of Colorado at Boulder Boulder, Colorado 80309

Journal de Théorie des Nombres de Bordeaux (2009)

- Volume: 21, Issue: 1, page 215-234
- ISSN: 1246-7405

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topHare, 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 $m < n$ 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 $m < n$ 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

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