The v-number of monomial ideals

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dc.contributor.author Saha, Kamalesh
dc.contributor.author Sengupta, Indranath
dc.coverage.spatial United Kingdom
dc.date.accessioned 2022-06-29T12:39:33Z
dc.date.available 2022-06-29T12:39:33Z
dc.date.issued 2022-06
dc.identifier.citation Saha, Kamalesh and Sengupta, Indranath, "The v-number of monomial ideals", Journal of Algebraic Combinatorics, DOI: 10.1007/s10801-022-01137-y, Jun. 2022. en_US
dc.identifier.issn 0925-9899
dc.identifier.issn 1572-9192
dc.identifier.uri https://doi.org/10.1007/s10801-022-01137-y
dc.identifier.uri https://repository.iitgn.ac.in/handle/123456789/7842
dc.description.abstract We show that the v-number of an arbitrary monomial ideal is bounded below by the v-number of its polarization and also find a criteria for the equality. By showing the additivity of associated primes of monomial ideals, we obtain the additivity of the v-numbers for arbitrary monomial ideals. We prove that the v-number v(I(G)) of the edge ideal I(G), the induced matching number im(G) and the regularity reg(R/I(G)) of a graph G, satisfy v(I(G))?im(G)?reg(R/I(G)), where G is either a bipartite graph, or a (C4,C5)-free vertex decomposable graph, or a whisker graph. There is an open problem in Jaramillo and Villarreal (J Combin Theory Ser A 177:105310, 2021), whether v(I)?reg(R/I)+1, for any square-free monomial ideal I. We show that v(I(G))>reg(R/I(G))+1, for a disconnected graph G. We derive some inequalities of v-numbers which may be helpful to answer the above problem for the case of connected graphs. We connect v(I(G)) with an invariant of the line graph L(G) of G. For a simple connected graph G, we show that reg(R/I(G)) can be arbitrarily larger than v(I(G)). Also, we try to see how the v-number is related to the Cohen-Macaulay property of square-free monomial ideals.
dc.description.statementofresponsibility by Kamalesh Saha and Indranath Sengupta
dc.language.iso en_US en_US
dc.publisher Springer en_US
dc.subject Polarization en_US
dc.subject Arbitrary en_US
dc.subject Bipartite graph en_US
dc.subject Line graph en_US
dc.subject Monomial ideals en_US
dc.subject Whisker graph en_US
dc.title The v-number of monomial ideals en_US
dc.type Article en_US
dc.relation.journal Journal of Algebraic Combinatorics


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