Universal lex ideal approximations of extended Hilbert functions and Hamilton numbers

JOURNAL OF ALGEBRA

卷:560

Article

Ananyan, Tigran¹  Hochster, Melvin² 

[1] Altair Engn, 1820 E Big Beaver Rd, Troy, MI 48083 USA

[2] Univ Michigan, Dept Math, Ann Arbor, MI 48109 USA

关键词: Polynomial ring;    Form;    Hilbert function;    Lex ideal;    Regular sequence;    Hamilton numbers;   

DOI  :  10.1016/j.jalgebra.2020.06.009

来源: Elsevier

【 摘 要 】

Let R-(h) denote the polynomial ring in variables x(1), ..., x(h) over a specified field K. We consider all of these rings simultaneously, and in each use lexicographic (lex) monomial order with x(1) > ... > xh. Given a fixed homogeneous ideal I in R-(h), for each d there is unique lex ideal generated in degree at most d whose Hilbert function agrees with the Hilbert function of I up to degree d. When we consider IR(N) for N >= h, the set B-d (I, N) of minimal generators for this lex ideal in degree at most d may change, but B-d(I, N) is constant for all N >> 0. We let B-d(I) denote the set of generators one obtains for all N >> 0, and we let b(d) = b(d) (I) be its cardinality. The sequences b1, ..., b(d), ... obtained in this way may grow very fast. Remarkably, even when I = (x(1)(2), x(2)(2)), one obtains a very interesting sequence, 0, 2, 3, 4, 6, 12, 924, 409620, .... This sequence is the same as Hd-1 + 1 for d >= 2, where H-d is the dth Hamilton number. The Hamilton numbers were studied by Hamilton and by Hammond and Sylvester because of their occurrence in a counting problem connected with the use of Tschirnhaus transformations in manipulating polynomial equations. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free