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.

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.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.1079
Classification : 11A07,  11A51
Mots clés : Factorising polynomials, congruence equations, Igusa conjecture
@article{JTNB_2019__31_1_255_0,
     author = {Hickman, Jonathan and Wright, James},
     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 = {http://archive.numdam.org/articles/10.5802/jtnb.1079/}
}
Hickman, Jonathan; Wright, James. 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. http://archive.numdam.org/articles/10.5802/jtnb.1079/

[1] Apostol, Tom M. Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer, 1976 | Zbl 0335.10001

[2] Bombieri, Enrico 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 | Article | Numdam | Zbl 0307.14011

[3] Cluckers, Raf Igusa and Denef–Sperber conjectures on nondegenerate p-adic exponential sums, Duke Math. J., Volume 141 (2008) no. 1, pp. 205-216 | Article | MR 2372152 | Zbl 1133.11048

[4] Cluckers, Raf Exponential sums: questions by Denef, Sperber, and Igusa, Trans. Am. Math. Soc., Volume 362 (2010) no. 7, pp. 3745-3756 | Article | MR 2601607 | Zbl 1204.11132

[5] Cluckers, Raf 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 | Article | MR 2861764 | Zbl 1231.42011

[6] Deligne, Pierre La conjecture de Weil. I, Publ. Math., Inst. Hautes Étud. Sci., Volume 43 (1974), pp. 273-307 | Article | Numdam | Zbl 0287.14001

[7] Denef, Jan 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 0749.11054

[8] Denef, Jan; Sperber, Steven Exponential sums mod p n and Newton polyhedra, Bull. Belg. Math. Soc. Simon Stevin (2001), pp. 55-63 | MR 1900398 | Zbl 1046.11057

[9] Eisenbud, David Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer, 1995 | Zbl 0819.13001

[10] Hickman, Jonathan; Wright, James An abstract L 2 Fourier restriction theorem (2018) (https://arxiv.org/abs/1801.03180) | Zbl 1420.43005

[11] Hickman, Jonathan; Wright, James The Fourier restriction and Kakeya problems over rings of integers modulo N, Discrete Anal., Volume 2018 (2018) no. 11, 11, 54 pages | MR 3819048 | Zbl 1404.43008

[12] Igusa, Jun-ichi Forms of higher degree, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 59, Tata Institute of Fundamental Research, 1978 | MR 546292 | Zbl 0417.10015

[13] Igusa, Jun-ichi An introduction to the theory of local zeta functions, AMS/IP Studies in Advanced Mathematics, 14, American Mathematical Society, 2000 | MR 1743467 | Zbl 0959.11047

[14] Kumar, Neeraj; Martino, Ivan Regular sequences of power sums and complete symmetric polynomials, Matematiche, Volume 67 (2012) no. 1, pp. 103-117 | MR 2927823 | Zbl 1246.05165

[15] Kunz, Ernst Introduction to commutative algebra and algebraic geometry, Modern Birkhäuser Classics, Birkhäuser/Springer, 1980 | Zbl 0432.13001

[16] Lachaud, Gilles; Rolland, Robert On the number of points of algebraic sets over finite fields, J. Pure Appl. Algebra, Volume 219 (2015) no. 11, pp. 5117-5136 | Article | MR 3351576 | Zbl 06448467

[17] Lang, Serge; Weil, André Number of points of varieties in finite fields, Am. J. Math., Volume 76 (1954), pp. 819-827 | Article | MR 65218 | Zbl 0058.27202

[18] Macdonald, Ian Grant Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, Clarendon Press, 1995 | Zbl 0824.05059

[19] Stepanov, Sergei A. 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 252400 | Zbl 0192.58002

[20] Wooley, Trevor D. A note on simultaneous congruences, J. Number Theory, Volume 58 (1996) no. 2, pp. 288-297 | Article | MR 1393617 | Zbl 0852.11017

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