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Sum of squares conjecture: the monomial case in \(\mathbb {C}^3\)

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Abstract

The goal of this article is to prove the Sum of Squares Conjecture for real polynomials \(r(z,\bar{z})\) on \(\mathbb {C}^3\) with diagonal coefficient matrix. This conjecture describes the possible values for the rank of \(r(z,\bar{z}) \left\Vert {z} \right\Vert ^2\) under the hypothesis that \(r(z,\bar{z})\left\Vert {z} \right\Vert ^2=\left\Vert {h(z)} \right\Vert ^2\) for some holomorphic polynomial mapping h. Our approach is to connect this problem to the degree estimates problem for proper holomorphic monomial mappings from the unit ball in \(\mathbb {C}^2\) to the unit ball in \(\mathbb {C}^k\). D’Angelo, Kos, and Riehl proved the sharp degree estimates theorem in this setting, and we give a new proof using techniques from commutative algebra. We then complete the proof of the Sum of Squares Conjecture in this case using similar algebraic techniques.

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Authors and Affiliations

  1. Department of Mathematics, Brigham Young University, Provo, UT, 84602, USA

    Jennifer Brooks

  2. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Dusty Grundmeier

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  1. Jennifer Brooks
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  2. Dusty Grundmeier
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Correspondence to Dusty Grundmeier.

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Brooks, J., Grundmeier, D. Sum of squares conjecture: the monomial case in \(\mathbb {C}^3\). Math. Z. 299, 919–940 (2021). https://doi.org/10.1007/s00209-021-02725-7

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