The power series Dedekind-Mertens number

Let (Formula presented.) and (Formula presented.) be the polynomial ring and the power series ring, respectively, over a commutative ring R with identity. For (Formula presented.), let A_f be the ideal of R generated by the coefficients of f. If g is a polynomial in (Formula presented.), then the Dedekind-Mertens number (Formula presented.) is the least positive integer k such that (Formula presented.) for every polynomial (Formula presented.). For a power series (Formula presented.), we define the power series Dedekind-Mertens number (Formula presented.) to be the least positive integer k (if any) such that (Formula presented.) for every power series (Formula presented.). Heinzer and Huneke showed that if (Formula presented.), then (Formula presented.) - (Formula presented.), where l- (Formula presented.) is the locally minimal number of generators of A_g. Surprisingly, for (Formula presented.), Epstein and Shapiro obtained a similar result that (Formula presented.) - (Formula presented.) provided that the ring R is Noetherian. They asked whether the same result still holds for an arbitrary ring Park, Kang, and Toan gave a negative answer to this question and further proved that if (Formula presented.), then (Formula presented.) - (Formula presented.), where l- (Formula presented.) is the locally minimal number of special generators of A_g. Note that we always have l- (Formula presented.) - (Formula presented.) for every power series (Formula presented.). In this paper, we show that there is a ring R with power series (Formula presented.) such that (Formula presented.) - (Formula presented.) while l- (Formula presented.) can be arbitrarily larger than l- (Formula presented.). In other words, the upper bound l- (Formula presented.) for (Formula presented.) is sharp and l- (Formula presented.) and (Formula presented.) are in general unrelated to the number l- (Formula presented.). This gives a complete answer to the question by Epstein and Shapiro.