Algebraic curves P(x)-Q(y)=0 and functional equations

Chebyshev constants, transfinite diameter, and computation on complex algebraic curves

Notions of directional Chebyshev constant and transfinite diameter have recently been studied on certain algebraic curves in $\mathbb{C}^2$. The theory is extended here to curves in $\mathbb{C}^N$ for arbitrary $N$. The results are analogous but require more methods from computational algebraic geometry.

Degeneracy of Holomorphic Curves into Algebraic Varieties

Applying the Second Main Theorem we deal with the algebraic degeneracy of entire holomorphic curves from the complex plane into a complex algebraic normal variety of positive log Kodaira dimension that admits a finite proper morphism to a semi-abelian variety. We will also discuss applications to the Kobayashi hyperbolicity problem.

An epsilon-delta bound for plane algebraic curves and its use for certified homotopy continuation of systems of plane algebraic curves

We explain how, given a plane algebraic curve $\mathcal{C}\colon f(x,y) = 0$, $x_1 \in \mathbb{C}$ not a singularity of $y$ w.r.t. $x$, and $\varepsilon > 0$, we can compute $δ> 0$ such that $|y_j(x_1) - y_j(x_2)| < \varepsilon$ for all holomorphic functions $y_j(x)$ which satisfy $f(x, y_j(x)) = 0$ in a neighbourhood of $x_1$ and for all $x_2$ with $|x_1 - x_2| < δ$. Consequently, we obtain an algorithm for reliable homotopy continuation of plane algebraic curves. As an example application, we study continuous deformation of closed discrete Darboux transforms. Moreover, we discuss a scheme fo…

吹胀-球面等价下复代数曲线的分类

Classification of complex algebraic curves under blow-spherical equivalence

本文在吹胀-球面等价框架下完整分类了复代数曲线，并证明次数在该等价下保持不变以及一个 Bernstein 型结果。

Algebraic approximations of fibrations in abelian varieties over a curve

For every fibration $f : X \to B$ with $X$ a compact Kähler manifold, $B$ a smooth projective curve, and a general fiber of $f$ an abelian variety, we prove that $f$ has an algebraic approximation.

次高斯实代数极大曲线的一个指数稀疏性结果

An Exponential Rarefaction Result for Sub-Gaussian Real Algebraic Maximal Curves

本文证明了与次高斯随机实全纯截面相关的实代数极大曲线具有指数稀有性，推广了Gayet和Welschinger在高斯情形下的已有结果。