平面代数曲线上Birkhoff插值问题研究
Research on the Birkhoff Interpolation Problem on Plane Algebraic Curves

本文以一元Birkhoff插值研究结果为基础，对二元Birkhoff插值泛函组的适定性问题进行了研究。通过提出弱Gr？bner基的概念及其发现其性质，提出了一种利用两条不同次数代数曲线相交的点，构造出二元Birkhoff插值问题适定泛函组的新方法，从而将该方法所得到的结果推广到一般情形。并得到了构造平面代数曲线二元Birkhoff插值适定泛函组的一般性方法和实用性较强的理论，最后给出了具体实验算例，对所得研究结论给予了验证。
This paper takes the research results of univariate Birkhoff interpolation as its foundation to study the stability problem of two-dimensional Birkhoff interpolation generalized function sets. By introducing the concept and properties of weak Gr？bner bases, a new method is obtained which utilizes the intersection of any two arbitrary algebraic curves to construct the two-dimensional Birkhoff well-posed interpolation functional systems. This method extends the research direction’s findings to general cases, providing a general approach and practical theory for constructing planar algebraic curve well-posed interpolation functional systems. Finally, specific experimental examples are provided to validate the research conclusions obtained.