Universal Lex Ideal Approximations of Extended Hilbert Functions and Hamilton Numbers

Abstract

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 withfor 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 b 1 , . . . , b d , . . . obtained in this way may grow very fast. Remarkably, even when I = (x 2 1 , x 2 2 ), one obtains a very interesting sequence, 0, 2, 3, 4, 6, 12, 924, 409620, . . .. This sequence is the same as H d−1 + 1 for d ≥ 2, where H d is the d th 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.