TY - JOUR
T1 - Numerical Characterizations for Integral Dependence of Graded Ideals
AU - Das, Suprajo
AU - Roy, Sudeshna
AU - Trivedi, Vijaylaxmi
N1 - Publisher Copyright:
© The Author(s) 2026. Published by Oxford University Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
PY - 2026/5
Y1 - 2026/5
N2 - Let (Formula presented) be an equidimensional standard graded Noetherian algebra over (Formula presented). Let (Formula presented) denote the Hilbert–Samuel multiplicity of (Formula presented). For any graded ideal (Formula presented) of (Formula presented), the Rees algebra (Formula presented) is a bigraded (Formula presented) -algebra. For any integer (Formula presented), we denote by (Formula presented) the (Formula presented) -diagonal subalgebra of (Formula presented), by (Formula presented) the (Formula presented) -multiplicity of (Formula presented), and by (Formula presented) the mixed multiplicities of (Formula presented). Let (Formula presented) be two non-nilpotent graded ideals in (Formula presented). Let (Formula presented) be the standard graded (Formula presented) -algebra (Formula presented), where (Formula presented) is an indeterminate with (Formula presented). Let (Formula presented) be the maximum of the generating degrees of both (Formula presented) and (Formula presented). Let (Formula presented) and (Formula presented). We show that (Formula presented) if and only if (Formula presented) for some (Formula presented). If (Formula presented) is a domain, then (Formula presented) and (Formula presented) for all (Formula presented) and (Formula presented) for some (Formula presented). Moreover, (Formula presented) if and only if the truncated ideals (Formula presented) and (Formula presented) share the same the polar multiplicity or the (Formula presented) -multiplicity or the (Formula presented) -multiplicity for some (Formula presented). The proofs of these results rely on the theory of density functions, which was developed in [15].
AB - Let (Formula presented) be an equidimensional standard graded Noetherian algebra over (Formula presented). Let (Formula presented) denote the Hilbert–Samuel multiplicity of (Formula presented). For any graded ideal (Formula presented) of (Formula presented), the Rees algebra (Formula presented) is a bigraded (Formula presented) -algebra. For any integer (Formula presented), we denote by (Formula presented) the (Formula presented) -diagonal subalgebra of (Formula presented), by (Formula presented) the (Formula presented) -multiplicity of (Formula presented), and by (Formula presented) the mixed multiplicities of (Formula presented). Let (Formula presented) be two non-nilpotent graded ideals in (Formula presented). Let (Formula presented) be the standard graded (Formula presented) -algebra (Formula presented), where (Formula presented) is an indeterminate with (Formula presented). Let (Formula presented) be the maximum of the generating degrees of both (Formula presented) and (Formula presented). Let (Formula presented) and (Formula presented). We show that (Formula presented) if and only if (Formula presented) for some (Formula presented). If (Formula presented) is a domain, then (Formula presented) and (Formula presented) for all (Formula presented) and (Formula presented) for some (Formula presented). Moreover, (Formula presented) if and only if the truncated ideals (Formula presented) and (Formula presented) share the same the polar multiplicity or the (Formula presented) -multiplicity or the (Formula presented) -multiplicity for some (Formula presented). The proofs of these results rely on the theory of density functions, which was developed in [15].
UR - https://www.scopus.com/pages/publications/105038387400
U2 - 10.1093/imrn/rnag088
DO - 10.1093/imrn/rnag088
M3 - Article
AN - SCOPUS:105038387400
SN - 1073-7928
VL - 2026
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 10
ER -