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The $ s$-multiplicity function of $ 2 \times 2$-determinantal rings


Authors: Lance Edward Miller and William D. Taylor
Journal: Proc. Amer. Math. Soc. 146 (2018), 2797-2810
MSC (2010): Primary 13D40; Secondary 05A15, 05A10
DOI: https://doi.org/10.1090/proc/13979
Published electronically: February 21, 2018
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Abstract: This article generalizes joint work of the first author and I. Swanson to the $ s$-multiplicity recently introduced by the second author. For $ k$ a field and $ X = [ x_{i,j}]$ an $ m \times n$-matrix of variables, we utilize Gröbner bases to give a closed form the length $ \lambda ( k[X] / (I_2(X) + \mathfrak{m}^{ \lceil sq \rceil } + \mathfrak{m}^{[q]} ))$, where $ s \in {\mathbf Z}[p^{-1}]$, $ q$ is a sufficiently large power of $ p$, and $ \mathfrak{m}$ is the homogeneous maximal ideal of $ k[X]$. This shows this length is always eventually a polynomial function of $ q$ for all $ s$.


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Additional Information

Lance Edward Miller
Affiliation: Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701
Email: lem016@uark.edu

William D. Taylor
Affiliation: Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701
Email: wdtaylor@uark.edu

DOI: https://doi.org/10.1090/proc/13979
Received by editor(s): August 19, 2017
Received by editor(s) in revised form: October 5, 2017, and October 8, 2017
Published electronically: February 21, 2018
Communicated by: Irena Peeva
Article copyright: © Copyright 2018 American Mathematical Society

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