Exploring Linear Relations Among Laurent Coefficients of Certain Hilbert Series

Abstract

In [Herbig-Herden-Seaton, arXiv:1605.01572 [math.CO] 2016], the authors considered rational functions of one variable, t, that satisfy a functional equation h(t). The function is in terms of integers a and d, where d is the pole order of h(t) at t=1. They found that depending on the value of r, where r=-(a+d), the coefficients of the Laurent expansion at t=1 satisfy various triangular linear relations, and so formed structures like that of Pascal's triangle or the Lucas triangle, for example. In this project, we experimentally investigate the extension of their findings, using a large collection of functions of two variables t1 and t2 that satisfy an analogous equation. We explore, using series expansions on Mathematica, whether the two variable functions are characterized by similar linear constraints (defined iteratively as the Laurent coefficients at t2 = 1 of the Laurent coefficients at t1 = 1). Our results suggest that there do seem to be similar relations, indicating a possible generalization to this case. Beyond the scope of this project, the motivation for studying these relations is as follows. By a theorem of R. Stanley, a graded Cohen-Macaulay domain A, where a is the a-invariant, is Gorenstein if and only if its Hilbert series satisfies the functional equation given above.