Mediterr. J. Math. (2019) 16:77 https://doi.org/10.1007/s00009-019-1337-7 c  Springer Nature Switzerland AG 2019 On the Normality of a Class of Monomial Ideals via the Newton Polyhedron Ibrahim Al-Ayyoub , Imad Jaradat and Khaldoun Al-Zoubi Abstract. Let I = 〈x a 1 1 ,...,x an n 〉⊂ R = K[x1,...,xn] with a1,...,an 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 〈x b 1 1 ,...,x bn n ,...,x br r 〉 ⊂ R[xn+1,...,xr ] is normal, where bi ∈{a1,...,an} for all i, this generalizes the work of Al-Ayyoub (Rocky Mt Math 39(1):1–9, 2009). If l = lcm(a1,...,an) and the integral clo- sure of  x a 1 1 ,...,x an n ,x l n+1  ⊂ R[xn+1] is not normal, then we show that the integral closure of 〈x a 1 1 ,...,x an n ,x s n+1 〉 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). Mathematics Subject Classification. 13B22, 06B10, 52B20. Keywords. Newton polyhedron, lattice points, integral closure, normal ideals, convex hull. 1. Introduction Let I be an ideal in a Noetherian ring R. The integral closure of I is the ideal I that consists of all elements of R that satisfy an equation of the form x n + d 1 x n−1 + ··· + d n−1 x + d n =0, d i ∈ I i (i =1,...,n). The ideal I is said to be integrally closed if I = I . An ideal is called normal if all of its powers are integrally closed. It is known that if R is a normal integral domain, then the Rees algebra R[It]= ⊕ n∈N I n t n is normal if and only if I is a normal ideal of R. This brings up the importance of normality of ideals as the Rees algebra is the algebraic counterpart of blowing up a scheme along a closed subscheme [12]. There is no concise solution to the problem of when a given ideal is normal, not even in the monomial case. A pleasing result of Reid, Roberts and Vitulli [9, Theorem 5.1] states that if the first n − 1 powers of a monomial ideal, in a polynomial ring of n variables over a field, are integrally closed, then the ideal is normal. 0123456789().: V,-vol