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Abstract
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We study the homological shifts of polymatroidal ideals. It is shown that the first homological shift ideal of any polymatroidal ideal is again polymatroidal, supporting a conjecture of Bandari, Bayati and Herzog that predicts that all homological shift ideals of a polymatroidal ideal are polymatroidal. We also study the “socle ideal” soc(I) of a polymatroidal ideal I and relate it to the highest homological shift ideal of I. It is shown that soc(I) is polymatroidal if: (i) I is a polymatroidal ideal generated in degree two, (ii) I is a polymatroidal ideal in at most three variables, (iii) I is matroidal, (iv) I is a principal Borel ideal, (v) I is a PLP–polymatroidal ideal, (vi) I is a LP–polymatroidal ideal.
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