Universal Lex Ideal Approximations of Extended Hilbert Functions and Hamilton Numbers

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 > ⋯ > x h . 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 I R ( 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 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 H d − 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.