Some recent transcendental techniques in algebraic and complex geometry

Transcendental holomorphic maps between real algebraic manifolds in a complex space

We give an example of a real algebraic manifold embedded in a complex space that does not satisfy the Nash-Artin approximation Property. This Nash-Artin approximation Property is closely related to the problem of determining when the biholomorphic equivalence for germs of real algebraic manifolds coincides with the algebraic equivalence. This example is an elliptic Bishop surface, and its construction is based on the functional equation satisfied by the generating series of some walks restricted to the quarter plane.

Some algebraic differential equations with few meromorphic solutions

A special class of autonomous algebraic differential equations is studied. No equations in the class have any entire transcendental solutions. In a sense, for almost all equations in the class, transcendental meromorphic solutions can also be excluded. With substantially fewer exceptional equations, transcendental solutions are either elliptic or of the form f(z) = g(exp(cz)) with g rational and c constant.

First order algebraic differential equations of genus zero

We utilise recent results about the transcendental solutions to Riccati differential equations to provide a comprehensive description of the nature of the transcendental solutions to algebraic first order differential equations of genus zero.

Kähler几何中的超越数值维与有理商

Transcendental Numerical Dimension and Rational Quotients in Kähler Geometry

该研究探究紧Kähler流形的超越数值维性质，建立其与纤维化退化除子的联系，结合Cao-Păun定理回答有理商相关问题，得到伪有效线丛的数值维不等式。

Residue Integrals and Waring's Formulas for a Class of Systems of Transcendental Equations in $\mathbb{C}^n$

The present article is focused on the study of a special class of systems of nonlinear transcendental equations for which classical algebraic and symbolic methods are inapplicable. For the purpose of the study of such systems, we develop a method for computing residue integrals with integration over certain cycles. We describe conditions under which the mentioned residue integrals coincide with power sums of the inverses to the roots of a system of equations (i.e., multidimensional Waring's formulas). As an application of the suggested method, we consider a problem of finding sums of multi-var…

Permutable entire functions satisfying algebraic differential equations

It is shown that if two transcendental entire functions permute, and if one of them satisfies an algebraic differential equation, then so does the other one.