The Poincaré Series of Modules over Generic Artinian Gorenstein Algebras of Even Socle Degree

Let Q = k[[x1, . . . , xn]] be the power series ring over a field k. Artinian Gorenstein quotients R = Q/I whose unique maximal ideal m satisfies ms 6= 0 = ms+1 are in correspondence via the Macaulay inverse system with degree s polynomials in n variables. Bøgvad constructed examples for which the Poincaré series of k over R is irrational. When s is even, we prove that such examples are rare. More precisely, if s is even and the inverse polynomial of R satisfies a generic condition, we prove that the Poincaré series of all finitely generated R-modules are rational, sharing a common denominator. We determine explicitly the denominator and the Poincaré series of k.