Abstract:A number of open problems in the field of several complex variables naturally lead to the study of bihomogeneous polynomials $r(z,\bar {z})$ on $\mathbb {C}^{n+1}$. In particular, both the Ebenfelt sum of squares conjecture and the degree estimate conjecture for proper rational mappings between balls in complex Euclidean spaces lead to the study of the rank of the bihomogeneous polynomial $r(z,\bar {z}) \left \lVert {z}\right \rVert ^2$ under certain additional hypotheses. When $r$ has a diagonal coefficient matrix, these questions reduce to questions about real homogeneous polynomials. More specifically, we are led to study the rank of $P=SQ$ when $Q$ is a homogeneous polynomial and $S(x) = \sum _{j=0}^n x_j$. In this paper, we use techniques from commutative algebra to estimate the minimum rank of $P=SQ$ under the additional hypothesis that $Q$ has maximum rank. The problem has already been solved for $n+1 \leq 3$, and so we consider $n+1\geq 4$. We obtain a minimum rank estimate that is sharp when $n+1=4$, and we exhibit a family of polynomials having this minimum rank. We also prove an estimate for $n+1>4$ that, while not sharp, is non-trivial.

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中文翻译：

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齐次多项式的秩问题

摘要：几个复变量领域的一些开放性问题自然而然地导致了对$\mathbb {C}^{n+1}$ 上的双齐次多项式$r(z,\bar {z})$ 的研究。特别是，Ebenfelt 平方和猜想和复杂欧几里德空间中球之间适当有理映射的度估计猜想导致对双齐次多项式 $r(z,\bar {z}) \left \ 的秩的研究lVert {z}\right \rVert ^2$ 在某些附加假设下。当 $r$ 有一个对角系数矩阵时，这些问题就归结为关于实齐次多项式的问题。更具体地说，当 $Q$ 是齐次多项式且 $S(x) = \sum _{j=0}^n x_j$ 时，我们研究了 $P=SQ$ 的秩。在本文中，我们使用交换代数的技术来估计 $P=SQ$ 在 $Q$ 具有最大秩的附加假设下的最小秩。$n+1\leq 3$ 的问题已经解决了，所以我们考虑$n+1\geq 4$。当 $n+1=4$ 时，我们获得了一个最小秩估计，并且我们展示了具有这个最小秩的多项式族。我们还证明了对 $n+1>4$ 的估计，虽然不尖锐，但并非无关紧要。

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