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.

A sharp bound is obtained for the number of ways to express the monomial ${X}^{n}$ as a product of linear factors over $ℤ/{p}^{\alpha }ℤ$. 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 $ℤ/{p}^{\alpha }ℤ$. 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.

Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1079
Classification: 11A07, 11A51
Keywords: Factorising polynomials, congruence equations, Igusa conjecture
Jonathan Hickman 1; James Wright 2

1 Mathematical Institute University of St Andrews North Haugh, St Andrews Fife, KY16 9SS, UK
2 Maxwell Institute of Mathematical Sciences and the School of Mathematics University of Edinburgh JCMB, King’s Buildings Peter Guthrie Tait Road Edinburgh, EH9 3FD, UK
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Jonathan Hickman; James Wright. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1079/

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