Functional transcendence via o-minimality (Q2825769)

The book deals with the Schanuel's conjecture and consists of 16 chapters. The first chapter contains basic concepts in algebraic independence theory. The second chapter deals with transcendence theory and introduces several famous theorems and conjectures. The third chapter presents the Schanuel's conjecture in the following way. Let \(n\) is a positive integer. Then for every complex numbers \(x_1,\dots ,x_n\) linearly independent over rational numbers we have \(\mathrm{tr.d.}_{\mathbb Q} \{x_1,\dots ,x_n,e^{x_1},\dots ,e^{x_n}\}\geq n\) where \(\mathrm{tr.d.}_{\mathbb Q} \mathbb S\) means transcendental degree of the set \(\mathbb S\) over the rational numbers; this is the cardinality of the transcendence basis over the rational numbers. Differential fields are defined in Chapter 4 and the topic of Chapter 5 is the Ax-Schanuel conjecture together with the theorem for differential fields. In the same spirit, the author formulates the Ax-Lindemann theorem as a similar result to the theorem of Lindemann concerning the linear independence of \(e^{z_1},\dots ,e^{z_n}\) in Chapter 6. Chapters 7, 8 and 9 deal with the Schanuel and Ax-Schanuel conjecture for modular forms. The modular Ax-Lindemann forms are the topic of Chapter 10. We can find the formulation of the Ax-Lindemann conjecture and the weak Ax-Schanuel conjecture in Chapter 11. Chapter 12 deals with the exponential Ax-Lindemann via o-minimality and Chapter 13 develops the modular Ax-Lindemann via o-minimality. A conjecture on intersections with tori, a uniform, weak and uniform weak Schanuel conjectures can be found in chapter 14 and there are also proof of some connections among them. Chapter 15 and 16 deal with the Zilber-Pink conjecture, the modular Schanuel conjecture, the Ax-Schanuel and the uniform modular Schanuel conjecture.NEWLINENEWLINEFor the entire collection see [Zbl 1326.11003].