On the Normality of a Class of Monomial Ideals via the Newton Polyhedron

Let I=〈x1a1,…,xnan〉⊂R=K[x1,…,xn] with a ₁ , … , a _n positive integers and K a field, and let J be the integral closure of I. A criterion for the normality of J is developed. This criterion is used to show that J is normal if and only if the integral closure of the ideal ⟨x1b1,…,xnbn,…,xrbr⟩⊂R[xn+1,…,xr] is normal, where b _i ∈ {a ₁ , … , a _n } for all i, this generalizes the work of Al-Ayyoub (Rocky Mt Math 39(1):1–9, 2009). If l= lcm(a ₁ , … , a _n ) and the integral closure of 〈x1a1,…,xnan,xn+1l〉⊂R[xn+1] is not normal, then we show that the integral closure of 〈x1a1,…,xnan,xn+1s〉 is not normal for any s> l. Also, we give a shorter proof of a main result of Coughlin (Classes of Normal Monomial Ideals. Ph.D. thesis, 2004).