Given a monic degree polynomial and a non-negative integer , we may form a new monic degree polynomial by raising each root of to the th power. We generalize a lemma of Dobrowolski to show that if and is prime then divides the resultant of and . We then consider the function . We show that for fixed and that this function is periodic in both and , and exhibits high levels of symmetry. Some discussion of its structure as a union of lattices is also given.
Étant donné un polynôme , unitaire de degré , et un entier positif , on peut définir un nouveau polynôme , unitaire de degré , en élevant chaque racine de à la puissance . Nous généralisons un lemme de Dobrowolski pour montrer que, si et est un nombre premier, alors divise le réesultant de et . Nous considérons alors la fonction . Nous montrons, pour et fixés, que cette fonction est périodique en et , et exhibons un grand nombre de symétries. Une étude de la structure comme réunion de réseaux est également faite.
DOI: 10.5802/jtnb.667
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@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/articles/10.5802/jtnb.667/} }
TY - JOUR TI - Patterns and periodicity in a family of resultants JO - Journal de Théorie des Nombres de Bordeaux PY - 2009 DA - 2009/// SP - 215 EP - 234 VL - 21 IS - 1 PB - Université Bordeaux 1 UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.667/ UR - https://zbmath.org/?q=an%3Apre05620678 UR - https://www.ams.org/mathscinet-getitem?mr=2537713 UR - https://doi.org/10.5802/jtnb.667 DO - 10.5802/jtnb.667 LA - en ID - JTNB_2009__21_1_215_0 ER -
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/
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