tailieunhanh - A note on unmixed ideals of Veronese bi-type

We classify the unmixed ideals of Veronese bi-type and in some cases we give a description of their associated prime ideals. In this paper some properties of these class of monomial ideals are discussed. In particular, our aim is to classify the unmixed Veronese bi-type ideals. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 1 – 7 ¨ ITAK ˙ c TUB doi: A note on unmixed ideals of Veronese bi-type Monica LA BARBIERA∗ University of Messina, Department of Mathematics Viale Ferdinando Stagno d’Alcontres, 31, 98166 Messina, Italy Received: • Accepted: • Published Online: • Printed: Abstract: We classify the unmixed ideals of Veronese bi-type and in some cases we give a description of their associated prime ideals. Key words: Unmixed ideals. Veronese bi-type ideals 1. Introduction Let R = K[X1 , . . . , Xn ; Y1 , . . . , Ym ] be a polynomial ring in two sets of variables over a field K . In recent papers, monomial ideals of R are introduced and their connection to bipartite complete graphs is studied ([4], [6]). In this paper we study a class of monomial ideals of R , so-called Veronese bi-type ideals. They are an extension of the ideals of Veronese type ([5]) in a polynomial ring in two sets of variables. More precisely, the ideals of Veronese bi-type are monomial ideals of R generated in the same degree: Lq,s = k+r=q Ik,s Jr,s , with n a a k, r ≥ 1 , where Ik,s is the Veronese-type ideal generated on degree k by the set {X1 i1 · · · Xnin | j=1 aij = 0 ≤ aij ≤ s, s ∈ {1, . . . , k}} and Jr,s is the Veronese-type ideal generated on degree r by the set b b {Y1 i1 · · · Ymim | m j=1 bij = r, 0 ≤ bij ≤ s, s ∈ {1, . . . , r}} ([2], [3]). For s = 2 the Veronese bi-type ideals are k, the ideals associated to bipartite graphs with loops ([2]). In this paper some properties of these class of monomial ideals are discussed. In particular, our aim is to classify the unmixed Veronese bi-type ideals. Establishing whenever an ideal is unmixed in general is a difficult problem because it is necessary to know all its associated prime ideals. In [8] equidimensional and unmixed ideals of Veronese type are characterized. Now we