平方和猜想：$\mathbb{C}^3$中的单式情形
Sum of Squares Conjecture: the Monomial Case in $\mathbb{C}^3$
摘要：
本文的目的是证明实多项式$r(z,\bar{z})$在具有对角系数矩阵的$\mathbb{C}^3$上的平方和猜想。这个猜想描述了在假设$r(z,\bar{z})\|z\|^2=\|h(z)\|^2$对某个全纯多项式映射$h$有秩的条件下，$r(z,\bar{z}) \|z\|^2$秩的可能值.我们的方法是将这个问题与从$\mathbb{C}^2$中的单位球到$\mathbb{C}^k$中的单位球的真全纯单性映射的度估计问题联系起来。D 'Angelo、Kos和Riehl在这种情况下证明了尖锐度估计定理，我们使用交换代数的技术给出了一个新的证明。然后，我们使用类似的代数技巧完成了这种情况下的平方和猜想的证明。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
