Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sum of Squares Conjecture: the Monomial Case in $\mathbb{C}^3$   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 $r(z,\bar{z})
\|z\|^2$ under the hypothesis that $r(z,\bar{z})\|z\|^2=\|h(z)\|^2$ 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.

1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
3
The categories of this paper are: physics
37
The categories of this paper are: supr-con
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
16
The categories of this paper are: stat
34
The categories of this paper are: chem-ph
19
The categories of this paper are: eess
27
The categories of this paper are: comp-gas
20
The categories of this paper are: acc-phys
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci