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Abstract
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Let S = K[x_1,...,x_n] be a polynomial ring over a field K and I \subset S be a monomial ideal with a linear resolution. Let m = (x_1,...,x_n) be the unique homogeneous maximal ideal and Im be a polymatroidal ideal. We prove that if either Im is polymatroidal with strong exchange property, or I is a monomial ideal in at most 4 variables, then I is polymatroidal. We also show that the first homological shift ideal of polymatroidal ideal is again polymatroidal.
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