代數幾何中，插值問題的目的在於尋找多項式，其零點依指定的重數通過給定的某些點。Alexander-Hirschowitz 定理給出了關於二重點上的插值問題的解答。在本篇論文中，我們將會探討由[Ch]與[BO]給出的Alexander-Hirschowitz定理的一個簡化後的證明，並以電腦驗證相關的結果。

The interpolation problem asks for the dimension of the linear system consisting of polynomials whose zero locus pass through a given collection of points with prescribed multiplicities. The Alexander-Hirschowitz theorem is the answer to the interpolation problem on general collections of double points. In this paper, we will review the simplified version of the proof of Alexander-Hirschowitz theorem given in [Ch] and [BO], explore the ideas behind the proof, and examine some of the results by computer.

1. Introduction (page 1)
2. Notations and Basic Properties (page 2)
3. Techniques Applied to Plane Curves (page 10)
4. Terracini’s Inductive Method (page 12)
5. The Degneration Argument (Horace’s Differential Method) (page 16)
6. The Cases of Cubics (page 26)
Reference (page 40)

[AH1] J. Alexander, A. Hirschowitz, An asymptotic vanishing theorem for generic unions of multiple points. Invent. Math. 2000, 140, 303-325
[AH2] J. Alexander, A. Hirschowitz, Polynomial interpolation in several variables, J. Alg. Geom. 4 1995, n.2, 201-222
[BO] M.C. Brambilla, ; G. Ottaviani, On the Alexander-Hirschowitz theorem. J. Pure Appl. Algebra 2008, 212, 1229-1251
[B] Bernardi et al., The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition, Mathematics, 6.12, 2018, 314
[Ch] K. Chandler, A brief proof of a maximal rank theorem for generic double points in projective space, Trans, Amer, Math. Soc. 353, 2001, no. 5, 1907-1920