超平面上多项式常数的度估计
Degree Estimates for Polynomials Constant on a Hyperplane
摘要：
研究复欧氏空间中球之间的真有理映射，自然会引出这种映射的嵌入维数与度之间的关系。单项式映射的特殊情况等价于本文所讨论的问题。估计多项式在$n$实变量中的次数$d$，假设具有非负系数并且在超平面上是常数，根据其项数$N$。当$n=1$。当$n=2$时，已知急剧结合的$d\le 2N-3$。本文包括两个主要结果。第一个为所有$n\ge 2$提供了一个界限，但对$n\ge 3$不明显。该界限意味着$n\ge 3$的更容易陈述的界限$d\le {4(2N-3)\over 3(2n-3)}$。第二个结果是一个稳定性定理;如果$n$足够大，给定$d$，那么尖锐界$d \le {N-1 \over n-1}$成立。在这种情况下，我们确定所有多项式的界限是尖锐的。
英文摘要：
The study of proper rational mappings between balls in complex Euclidean spaces naturally leads to the relationship between the degree and imbedding dimension of such a mapping. The special case for monomial mappings is equivalent to the question discussed in this paper. Estimate the degree $d$ of a polynomial in $n$ real variables, assumed to have non-negative coefficients and to be constant on a hyperplane, in terms of the number $N$ of its terms. No such estimate is possible when $n=1$. The sharp bound $d\le 2N-3$ is known when $n=2$. This paper includes two main results. The first provides a bound, not sharp for $n\ge 3$, for all $n\ge 2$. This bound implies the more easily stated bound $d\le {4(2N-3)\over 3(2n-3)}$ for $n\ge 3$. The second result is a stabilization theorem; if $n$ is sufficiently large given $d$, then the sharp bound $d \le {N-1 \over n-1}$ holds. In this situation we determine all polynomials for which the bound is sharp.
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。
