Patterns and periodicity in a family of resultants

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

doi:10.5802/jtnb.667

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, Volume 21 (2009) no. 1, pp. 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

DOI: 10.5802/jtnb.667

Author's affiliations:

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

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Kevin G. Hare; David McKinnon; Christopher D. Sinclair. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.667/

References
Cited by

[1] E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial. Acta. Arith. 34 (1979), 391–401. | MR | Zbl

[2] David S. Dummit, Richard M. Foote, Abstract algebra. Prentice Hall Inc., Englewood Cliffs, NJ, 1991. | MR | Zbl

[3] Rudolf Lidl, Harald Niederreiter, Introduction to finite fields and their applications. Cambridge University Press, Cambridge, 1986. | MR | Zbl

[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. | MR | Zbl

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