COMPUTING THE SPREADING AND COVERING NUMBERS

Let S = k[x ₁,…,x _n ], d a positive integer, and suppose that S _D is the vector space of all polynomials of degree d in S. Define α _n (d) ? max { dim _k V| V monomial subspace of S _d , dim _k S ₁ V = n dim _k V} and ρ _n (d +1) ? min {dim _k V | V monomial subspace of S _d , S ₁ V = S _d+1}. The numbers α _n (d) and ρ _n (d+ 1) are called the spreading numbers and covering numbers, respectively. We describe an approach to calculate these numbers that uses simplicial complexes.