On the Betti polynomials of certain graded ideals

Let [Formula omitted.] be a polynomial ring over a field K and I be a nonzero graded ideal of S. Then, for t[Gt]0, the Betti number [Formula omitted.] is a polynomial in t, which is denoted by [Formula omitted.] . It is proved that [Formula omitted.] is vanished or of degree [script small l](I)-1 pr...

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Bibliographic Details
Published in:Communications in algebra 2018-07, Vol.46 (7), p.3135-3146
Main Authors: Failla, Gioia, Tang, Zhongming
Format: Article
Language:English
Subjects:
Lower bounds
Mathematical analysis
Polynomials
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Summary:Let [Formula omitted.] be a polynomial ring over a field K and I be a nonzero graded ideal of S. Then, for t[Gt]0, the Betti number [Formula omitted.] is a polynomial in t, which is denoted by [Formula omitted.] . It is proved that [Formula omitted.] is vanished or of degree [script small l](I)-1 provided I is a monomial ideal generated in a single degree or grade(&#[MATHEMATICAL FRAKTUR SMALL L];R(I)) = codim(&#[MATHEMATICAL FRAKTUR SMALL L];R(I)) where [Formula omitted.] and R(I) is the Rees ring of I. One lower bound for the leading coefficient of [Formula omitted.] is given. When I is a Borel principal monomial ideal, [Formula omitted.] is calculated explicitly.
ISSN:0092-7872
1532-4125
DOI:10.1080/00927872.2017.1404077