On the h-vector of a simplicial complex with Serre’s condition

Let Δ be a (d−1)-dimensional simplicial complex and let h(Δ)=(h0,h1,…,hd) be its h-vector. A recent result of Murai and Terai guarantees that if Δ satisfies Serre’s condition (Sr), then (h0,h1,…,hr) is an M-vector and hr+hr+1+⋯+hd is nonnegative. In this article, we extend the result of Murai and Terai by giving r extra necessary conditions. More precisely, we prove that if Δ satisfies Serre’s condition (Sr), then , 0≤i≤r≤d, are all nonnegative.