有理映射的均值猜想
Mean Value Conjectures for Rational Maps
摘要：
设p为单复变数多项式。Smale均值猜想估计|p'（z）|根据从（z，p（z））到$p$图上某个驻点的弦的梯度。这个猜想并没有立即推广到有理映射，因为它的公式在仿射映射群下是不变的，而不是在完整的莫比乌斯群下。在这里，我们给出了两个可能的推广理性映射，这两个都是莫比乌斯不变量。在这两种情况下，我们证明了一个版本，一个较弱的常数，在平行的Smale的平均值猜想的情况。最后，我们讨论了一些候选的极值有理映射，即所有临界点都是不动点的有理映射。
英文摘要：
Let p be a polynomial in one complex variable. Smale's mean value conjecture estimates |p'(z)| in terms of the gradient of a chord from (z, p(z)) to some stationary point on the graph of $p$. The conjecture does not immediately generalise to rational maps since its formulation is invariant under the group of affine maps, not the full Mobius group. Here we give two possible generalisations to rational maps, both of which are Mobius invariant. In both cases we prove a version with a weaker constant, in parallel to the situation for Smale's mean value conjecture. Finally, we discuss some candidate extremal rational maps, namely rational maps all of whose critical points are fixed points.
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