A combinatorial proof of Gotzmann's persistence theorem for monomial ideals

Gotzmann proved the persistence for minimal growth for ideals. His theorem is called Gotzmann’s persistence theorem. In this paper, based on the combinatorics on binomial coefficients, a simple combinatorial proof of Gotzmann’s persistence theorem in the special case of monomial ideals is given. Introduction Let K be an arbitrary field, R = K[x1, x2, . . . , xn] the polynomial ring with deg(xi) = 1 for i = 1, 2, . . . , n. Let M denote the set of variables {x1, x2, . . . , xn}, M the set of all monomials of degree d, where M = {1} , and Mi = M \ {xi}. For a monomial u ∈ R and for a subset V ⊂ M, we define uV = {uv|v ∈ V } and MV = {xiv|v ∈ V, i = 1, 2, . . . , n}. For a finite set V ⊂ M , we write |V | for the number of the elements of V . Let gcd(V ) denote the greatest common divisor of the monomials belonging to V . Let n and h be positive integers. Then h can be written uniquely in the form, called the nth binomial representation of h, h = ( h(n) + n n ) + ( h(n− 1) + n− 1 n− 1 ) + · · ·+ ( h(i) + i i ) , where h(n) ≥ h(n− 1) ≥ · · · ≥ h(i) ≥ 0, i ≥ 1. See [3, Lemma 4.2.6]. Let ( h(1) s(1) )