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An application of Macaulay’s estimate to sums of squares problems in several complex variables

Authors: Dusty Grundmeier and Jennifer Halfpap Kacmarcik
Journal: Proc. Amer. Math. Soc. 143 (2015), 1411-1422
MSC (2010): Primary 13D40, 32A17, 32H99
Published electronically: December 9, 2014
MathSciNet review: 3314056
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Abstract: Several questions in complex analysis lead naturally to the study of bihomogeneous polynomials $r(z,\bar {z})$ on $\mathbb {C}^n \times \mathbb {C}^n$ for which $r(z,\bar {z})\left \lVert z \right \rVert ^{2d}=\left \lVert h(z) \right \rVert ^2$ for some natural number $d$ and a holomorphic polynomial mapping $h=(h_1, \ldots , h_K)$ from $\mathbb {C}^n$ to $\mathbb {C}^K$. When $r$ has this property for some $d$, one seeks relationships between $d$, $K$, and the signature and rank of the coefficient matrix of $r$. In this paper, we reformulate this basic question as a question about the growth of the Hilbert function of a homogeneous ideal in $\mathbb {C}[z_1,\ldots ,z_n]$ and apply a well-known result of Macaulay to estimate some natural quantities.

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Additional Information

Dusty Grundmeier
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Address at time of publication: Department of Mathematical Sciences, Ball State University, Muncie, Indiana 47306

Jennifer Halfpap Kacmarcik
Affiliation: Department of Mathematical Sciences, University of Montana, Missoula, Montana 59812

Received by editor(s): March 31, 2013
Received by editor(s) in revised form: August 2, 2013
Published electronically: December 9, 2014
Additional Notes: The first author was partially supported by NSF RTG grant DMS-1045119.
The second author was supported in part by NSF grant DMS 1200815.
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.