Unique range sets and uniqueness polynomials in positive characteristic II
Unique range sets and uniqueness polynomials in positive characteristic II
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正特征 II 中的唯一值域集和唯一性多项式
DOI：
10.4064/aa116-2-2
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发表时间：
2003
期刊：
影响因子：
0.7
通讯作者：
P. Wong
中科院分区：
文献类型：
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作者：
T. An;Julietzu;P. Wong
如果E(f， S) = E(g， S)，两个函数f和g (f)被称为共享S，计算多重性。对于F，如果条件E(F， S) = E（g， S）对于F， g∈F意味着F≡g，则集合S称为计数多重的唯一值域集。定义在K上的多项式P称为F的唯一多项式，如果条件P (F) = P (g)对于F， g∈F意味着F≡g；如果条件P (f) = cP (g)对于f， g∈f和某个非零常数c意味着c = 1且f≡g，则P称为强唯一性多项式。在[1]中，我们证明了在正特征的情况下，一组特殊的多项式是非阿基米德亚纯函数的强唯一性多项式。这是通过显式构造P2中与特殊族相关的曲线的朗斯基型正则1-型(s)来实现的。由非阿基米德均匀化定理可知，这些曲线是非阿基米德双曲曲线，即不存在非常数的非阿基米德解析映射到曲线中。在处理比[1]中所考虑的更一般形式的多项式时，我们无法明确地在相关曲线上构造正则1-形式(s)，即正则束KC的正则截面；然而，我们能够构造出显规则的1-形式(s)的m-褶对称积，即正则束Km C的幂的正则部分，这仍然意味着
Two functions f and g of F are said to share S, counting multiplicity, if E(f, S) = E(g, S). A set S is called a unique range set , counting multiplicity, for F , if the condition E(f, S) = E(g, S) for f, g ∈ F implies that f ≡ g. A polynomial P defined over K is called a uniqueness polynomial for F if the condition P (f) = P (g) for f, g ∈ F implies that f ≡ g; P is called a strong uniqueness polynomial if the condition P (f) = cP (g) for f, g ∈ F and some non-zero constant c implies that c = 1 and f ≡ g. In [1] we showed, in the case of positive characteristic, that a special family of polynomials are strong uniqueness polynomials for non-archimedean meromorphic functions. This was accomplished by explicitly constructing, for the curves in P2 associated to the special family, regular 1-form(s) of Wronskian type. It then follows from the non-archimedean uniformization theorem that these curves are non-archimedean hyperbolic, i.e., there is no non-constant non-archimedean analytic map into the curves. In dealing with more general forms of polynomials than those considered in [1] we are unable to explicitly construct regular 1-form(s), i.e., regular sections of the canonical bundle KC , on the associated curves; however, we are able to construct explicitly regular m-fold symmetric product of 1-form(s), i.e., regular sections of powers of the canonical bundle Km C , and this still implies that
H. Fujimoto：“论共享有限集的亚纯函数的唯一性”Amer。
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