Polytopal resolutions of dense ideals

Let S = k[x₁,⋯,xₙ] and I=(x₁,⋯,xₙ)d. In this document, we study ideals obtained by removing small number of generators from G(I). We introduce a minimal resolution for I, the ideal generated by G(I) ∖ {m}, where m is a generator of I. We also give an upper bound for the Hirsch diameter of the graph associated with J where J is an equigenerated monomial ideal with linear presentation.