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.