Counting factorisations of monomials over rings of integers modulo N
Jonathan Hickman1 ; James Wright2
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
Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 1, pp. 255-282.

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 α 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.

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 α 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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1079
Classification : 11A07, 11A51
Mots clés : 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
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{JTNB_2019__31_1_255_0,
     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},
     pages = {255--282},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {1},
     year = {2019},
     doi = {10.5802/jtnb.1079},
     mrnumber = {3994730},
     zbl = {07246524},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1079/}
}
TY  - JOUR
AU  - Jonathan Hickman
AU  - James Wright
TI  - Counting factorisations of monomials over rings of integers modulo $N$
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2019
SP  - 255
EP  - 282
VL  - 31
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1079/
DO  - 10.5802/jtnb.1079
LA  - en
ID  - JTNB_2019__31_1_255_0
ER  - 
%0 Journal Article
%A Jonathan Hickman
%A James Wright
%T Counting factorisations of monomials over rings of integers modulo $N$
%J Journal de théorie des nombres de Bordeaux
%D 2019
%P 255-282
%V 31
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1079/
%R 10.5802/jtnb.1079
%G en
%F JTNB_2019__31_1_255_0
Jonathan Hickman; James Wright. Counting factorisations of monomials over rings of integers modulo $N$. Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 1, pp. 255-282. doi : 10.5802/jtnb.1079. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1079/

[1] Tom M. Apostol Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer, 1976 | Zbl

[2] Enrico Bombieri Counting points on curves over finite fields (d’après S. A. Stepanov), Séminaire Bourbaki, 25ème année (1972/1973) (Lecture Notes in Mathematics), Springer, 1974, pp. 234-241 | DOI | Numdam | Zbl

[3] Raf Cluckers Igusa and Denef–Sperber conjectures on nondegenerate p-adic exponential sums, Duke Math. J., Volume 141 (2008) no. 1, pp. 205-216 | DOI | MR | Zbl

[4] Raf Cluckers Exponential sums: questions by Denef, Sperber, and Igusa, Trans. Am. Math. Soc., Volume 362 (2010) no. 7, pp. 3745-3756 | DOI | MR | Zbl

[5] Raf Cluckers Analytic van der Corput lemma for p-adic and F q ((t)) oscillatory integrals, singular Fourier transforms, and restriction theorems, Expo. Math., Volume 29 (2011) no. 4, pp. 371-386 | DOI | MR | Zbl

[6] Pierre Deligne La conjecture de Weil. I, Publ. Math., Inst. Hautes Étud. Sci., Volume 43 (1974), pp. 273-307 | DOI | Numdam | Zbl

[7] Jan Denef Report on Igusa’s local zeta function, Séminaire Bourbaki. Volume 1990/91 (Astérisque), Volume 201-203, Société Mathématique de France, 1991, pp. 201-203 | Numdam | Zbl

[8] Jan Denef; Steven Sperber Exponential sums mod p n and Newton polyhedra, Bull. Belg. Math. Soc. Simon Stevin (2001), pp. 55-63 | MR | Zbl

[9] David Eisenbud Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer, 1995 | Zbl

[10] Jonathan Hickman; James Wright An abstract L 2 Fourier restriction theorem (2018) (https://arxiv.org/abs/1801.03180) | Zbl

[11] Jonathan Hickman; James Wright The Fourier restriction and Kakeya problems over rings of integers modulo N, Discrete Anal., Volume 2018 (2018) no. 11, 11, 54 pages | MR | Zbl

[12] Jun-ichi Igusa Forms of higher degree, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 59, Tata Institute of Fundamental Research, 1978 | MR | Zbl

[13] Jun-ichi Igusa An introduction to the theory of local zeta functions, AMS/IP Studies in Advanced Mathematics, 14, American Mathematical Society, 2000 | MR | Zbl

[14] Neeraj Kumar; Ivan Martino Regular sequences of power sums and complete symmetric polynomials, Matematiche, Volume 67 (2012) no. 1, pp. 103-117 | MR | Zbl

[15] Ernst Kunz Introduction to commutative algebra and algebraic geometry, Modern Birkhäuser Classics, Birkhäuser/Springer, 1980 | Zbl

[16] Gilles Lachaud; Robert Rolland On the number of points of algebraic sets over finite fields, J. Pure Appl. Algebra, Volume 219 (2015) no. 11, pp. 5117-5136 | DOI | MR | Zbl

[17] Serge Lang; André Weil Number of points of varieties in finite fields, Am. J. Math., Volume 76 (1954), pp. 819-827 | DOI | MR | Zbl

[18] Ian Grant Macdonald Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, Clarendon Press, 1995 | Zbl

[19] Sergei A. Stepanov The number of points of a hyperelliptic curve over a finite prime field, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 33 (1969), pp. 1171-1181 | MR | Zbl

[20] Trevor D. Wooley A note on simultaneous congruences, J. Number Theory, Volume 58 (1996) no. 2, pp. 288-297 | DOI | MR | Zbl

[21] James Wright Exponential sums and polynomial congruences in two variables: the quasi-homogeneous case (2012) (https://arxiv.org/abs/1202.2686)

Cité par Sources :