Around Exponential-Algebraic Closedness

F. Gallinaro
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引用次数: 1

Abstract

Abstract We present some results related to Zilber’s Exponential-Algebraic Closedness Conjecture, showing that various systems of equations involving algebraic operations and certain analytic functions admit solutions in the complex numbers. These results are inspired by Zilber’s theorems on raising to powers. We show that algebraic varieties which split as a product of a linear subspace of an additive group and an algebraic subvariety of a multiplicative group intersect the graph of the exponential function, provided that they satisfy Zilber’s freeness and rotundity conditions, using techniques from tropical geometry. We then move on to prove a similar theorem, establishing that varieties which split as a product of a linear subspace and a subvariety of an abelian variety A intersect the graph of the exponential map of A (again under the analogues of the freeness and rotundity conditions). The proof uses homology and cohomology of manifolds. Finally, we show that the graph of the modular j-function intersects varieties which satisfy freeness and broadness and split as a product of a Möbius subvariety of a power of the upper-half plane and a complex algebraic variety, using Ratner’s orbit closure theorem to study the images under j of Möbius varieties. Abstract prepared by Francesco Paolo Gallinaro E-mail: francesco.gallinaro@mathematik.uni-freiburg.de URL: https://etheses.whiterose.ac.uk/31077/
围绕指数代数封闭性
摘要本文给出了关于Zilber指数-代数闭性猜想的一些结果,证明了各种涉及代数运算的方程组和某些解析函数在复数中有解。这些结果的灵感来自于齐尔伯关于幂次幂的定理。我们使用热带几何的技术证明了作为可加群的线性子空间和乘法群的代数子空间的乘积分裂的代数变体相交于指数函数的图,只要它们满足Zilber的自由和圆度条件。然后我们继续证明一个类似的定理,建立作为线性子空间和阿贝尔变体a的子变体的乘积分裂的变体与a的指数映射的图相交(再次在自由和圆度条件的类似情况下)。证明使用流形的同调和上同调。最后,我们利用Ratner的轨道闭包定理研究了Möbius种j下的像,证明了模j函数的图与满足自由度和宽度的变种相交并分裂为上半平面幂次的Möbius子变种与复代数变种的乘积。作者:Francesco Paolo Gallinaro E-mail: francesco.gallinaro@mathematik.uni-freiburg.de URL: https://etheses.whiterose.ac.uk/31077/
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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