Universal Lex Ideal Approximations of Extended Hilbert Functions and Hamilton Numbers

Let $R^h$ denote the polynomial ring in variables $x_1,\,\ldots,\, x_h$ over a specified field $K$. We consider all of these rings simultaneously, and in each use lexicographic (lex) monomial order with $x_1 > \cdots > x_h$. 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 $IR^N$ for $N \geq h$, the set $\mathfrak{B}_d(I,N)$ of minimal generators for this lex ideal in degree at most $d$ may change, but $\mathfrak{B}_d(I,N)$ is constant for all $N \gg 0$. We let $\mathfrak{B}_d(I)$ denote the set of generators one obtains for all $N \gg 0$, and we let $b_d = b_d(I)$ be its cardinality. The sequences $b_1, \, \ldots, \, b_d, \, \ldots$ 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,$\,\ldots$. This sequence is the same as $H_{d-1} + 1$ for $d \geq 2$, 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.