The analytic spread of codimension two monomial varieties

Let ${\rm} A=k{u_{1}^{a_{1}}},{u_{2}^{a_{2}}},\dots,{u_{n}^{a_{n}}},{u_{1}^{c_{1}}} \dots {u_{n}^{c_{n}}},{u_{1}^{b_{1}}} \dots {u_{n}^{b_{n}}}]\ \subset k{u_{1}}, \dots {u_{n}}],$ where, a_j, b_j, C_j ∈ ?, a_j > 0, (b_j, C_j) ≠ (0,0) for 1 ≤ j ≤ n, and, further ${\underline b}:=\ ({b_{1}}, \dots,{b_{n}})\ \not=\ 0 $ and ${\underline c}:=\ ({c_{1}}, \dots,{c_{n}})\ \not=\ 0 $ . The main result says that the defining ideal I ? m = (x₁,…, x_n, y, z) ? kx₁,…, x_n, y, z] of the semigroup ring A has analytic spread ?(I_m) at most three.