Generating functions for borders
Jan Snellman
TL;DR
The paper derives explicit generating functions for the index of lattice points relative to a finite order ideal, connecting border bases with a combinatorial generating-function framework. It proves that the index satisfies a tropical-type difference equation and shows that the generating function $Ind_O(y_1,\dots,y_n)$ is rational, with a closed form in dimension two and a general $P_n$-based formula in higher dimensions. The central technical tool is an admissible decomposition of the complement $T([n])\setminus O$ and a universal formula for multivariate linear-forms generating functions. The results give explicit, computable expressions for the index generating function and reveal its polynomial nature after multiplying by $\prod (1-y_i)^2$.
Abstract
We give the generating function for the index of integer lattice points, relative to a finite order ideal. The index is an important concept in the theory of border bases, an alternative to Gröbner bases. Equivalently, we explicitly solve a class of difference equations where the right-hand side is the minimum of a number of affine forms.