论文标题
椭圆函数字段上的代数独立性和差异方程
Algebraic independence and difference equations over elliptic function fields
论文作者
论文摘要
对于晶格λin，将k_λ为λ-椭圆函数的场。对于大于1的两个相对质数整数P（分别为Q），请考虑通过P（resp。Q）在椭圆形曲线\ Mathbb {C}/λ上乘以p（resp。Q）给出的K_λ的内态性。我们证明，如果f（分别g）是复杂的劳伦功率序列，可满足K_λ相对于ϕ（分别ψ）的线性差异方程，则有二分法。要么，对于λ的某些sublaticeλ'，f或g之一属于ring k_ {λ'} [z，z，z^{ - 1}，ζ（z，λ'）]，其中ζ（z，λ'）是weiersstrass zeta zeta zeta函数，或f和f和f和f和f与k_λ独立。这是Adamczewski，Dreyfus，Hardouin和Wibmer（在理性函数领域）的近期定理的椭圆类似物。
For a lattice Λin the complex plane, let K_Λ be the field of Λ-elliptic functions. For two relatively prime integers p (respectively q) greater than 1, consider the endomorphisms ψ(resp. ϕ) of K_Λ given by multiplication by p (resp. q) on the elliptic curve \mathbb{C}/Λ. We prove that if f (resp. g) are complex Laurent power series that satisfy linear difference equations over K_Λ with respect to ϕ(resp. ψ) then there is a dichotomy. Either, for some sublattice Λ' of Λ, one of f or g belongs to the ring K_{Λ'}[z,z^{-1},ζ(z,Λ')], where ζ(z,Λ') is the Weierstrass zeta function, or f and g are algebraically independent over K_Λ. This is an elliptic analogue of a recent theorem of Adamczewski, Dreyfus, Hardouin and Wibmer (over the field of rational functions).
