Algebra Colloquium c ⃝ 2018 AMSS CAS & SUZHOU UNIV Algebra Colloquium 25 : 1 (2018) 71–80 DOI: 10.1142/S1005386718000056 On Non-standard Hilbert Functions ∗ Amir Bagheri Marand Technical College, University of Tabriz, Tabriz, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM) P. O. Box: 19395-5746, Tehran, Iran E-mail: a bageri@tabrizu.ac.ir Rahim Rahmati-Asghar Department of Mathematics, Faculty of Basic Sciences University of Maragheh, P. O. Box: 55181-83111, Maragheh, Iran E-mail: rahmatiasghar.r@maragheh.ac.ir Received 20 April 2017 Revised 11 September 2017 Communicated by Nanqing Ding Abstract. Let S = k[x1,...,xn] be a non-standard polynomial ring over a field k and let M be a finitely generated graded S-module. In this paper, we investigate the behaviour of Hilbert function of M and its relations with lattice point counting. More precisely, by using combinatorial tools, we prove that there exists a polytope such that the image of Hilbert function in some degree is equal to the number of lattice points of this polytope. 2010 Mathematics Subject Classification: 13A30, 13A02, 13D40 Keywords: polynomial ring, non-standard Hilbert function, Hermite normal form 1 Introduction Let k be a field and S = k[x 1 ,...,x n ] be a graded polynomial ring. For a finitely generated graded S-module M , the Hilbert function of M is a mapping from the set of degrees to Z. By this map, every degree is sent to the dimension of the homogeneous component of M at that degree. In fact, one can say that the Hilbert function of M measures the growth of dimension of the homogeneous components of M . The concept of Hilbert function is an important topic in graded commutative algebra, and it has many relations with other invariants of this field. Note that the computation of Hilbert function is not easy in general. There are some results on the behaviour of Hilbert functions in special cases such as monomial ideals, but these techniques do not work in general for arbitrary modules. * The research of the first author was supported in part by a grant from IPM (No. 94130024). Algebra Colloq. 2018.25:71-80. Downloaded from www.worldscientific.com by 107.175.205.117 on 07/15/19. Re-use and distribution is strictly not permitted, except for Open Access articles.