Counting factorisations of monomials over rings of integers modulo N
Jonathan Hickman; James Wright
Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 1, p. 255-282

A sharp bound is obtained for the number of ways to express the monomial X n as a product of linear factors over /p α . The proof relies on an induction-on-scale procedure which is used to estimate the number of solutions to a certain system of polynomial congruences. The method also applies to more general systems of polynomial congruences that satisfy a non-degeneracy hypothesis.

Dans cet article, on obtient une majoration optimale du nombre de façons d’écrire le monôme X n comme produit de facteurs linéaires sur /p α . La démonstration utilise une récurrence pour estimer le nombre de solutions d’un certain système de congruences polynomiales. La méthode s’applique également aux systèmes de congruences polynomiales plus généraux qui satisfont une hypothèse de non-dégénérescence.

Received : 2018-08-18
Revised : 2019-02-18
Accepted : 2019-04-08
Published online : 2019-07-29
DOI : https://doi.org/10.5802/jtnb.1079
Classification:  11A07,  11A51
Keywords: Factorising polynomials, congruence equations, Igusa conjecture
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     author = {Jonathan Hickman and James Wright},
     title = {Counting factorisations of monomials over rings of integers modulo $N$},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {1},
     year = {2019},
     pages = {255-282},
     doi = {10.5802/jtnb.1079},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2019__31_1_255_0}
}
Hickman, Jonathan; Wright, James. Counting factorisations of monomials over rings of integers modulo $N$. Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 1, pp. 255-282. doi : 10.5802/jtnb.1079. jtnb.centre-mersenne.org/item/JTNB_2019__31_1_255_0/

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