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Abstract
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Let R = K[x_1, ..., x_n] be the polynomial ring in n variables over a field K and I be a monomial ideal generated in degree d. Bandari and Herzog conjectured that a monomial ideal I is polymatroidal if and only if all its monomial localizations have a linear resolution. In this paper we give an affirmative answer to the conjecture in the following cases: (i) height(I) = n − 1; (ii) I contains at least n − 3 pure powers of the variables x_1^d , ..., x_n-3^d; (iii) I is a monomial ideal in at most four variables
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