On the Hilbert function of the tangent cone of a monomial curve

In this paper we study theHilbert function of grm(R),when R is a numerical semigroup ring or, equivalently, the coordinate ring of a monomial curve. In particular, we prove a sufficient condition for a numerical semigroup ring in order get a non-decreasing Hilbert function, without making any assumption on its embedding dimension; moreover, we show how this new condition allows us to improve known results about this problem. To this end we use certain invariants of the semigroup, with particular regard to its Apéry-set.