若方程式 f(x,y) = a0(x)+a1(x)*y+a2(x)*y2+...+an(x)*yn = 0, ai(x)∈C(x)∗, 我們要找出解 y(x) = x^{r1}(c1 + x^{r2}(c2 + x^{r3}(c3 + ...))), r2,r3,r4,...> 0, 並討論 y(x) 分支的情形以及何時會出現 {r1,r2,r3,...} 的最小公分母, 最後再算 y(x) 的收斂範圍。

If we have an equation that is f(x,y) = a0(x)+a1(x)*y+a2(x)*y2+...+an(x)*yn = 0, ai(x)∈C(x)∗, we want to find solutions which are of the form x^{r1}(c1 + x^{r2}(c2 + x^{r3}(c3 + ...))), r2,r3,r4,...> 0, and we will discuss the bifurcation of y(x) and when the lowest common denominator of {r1,r2,r3,...} appears. Finally, we compute the range of convergence of y(x) expansion.

1 由簡單的例題入手 p01
2 y(x) 的分支      p08
3 r 的最小公分母   p08
4 一般情況的例題   p10
5 y(x) 的收斂範圍  p14
6 總結             p15
參考文獻           p17

參考文獻
[1] D. N. Berstein (1975), “The number of roots of a system of equations”, Func-tional Anal. Appl., 9, 1–4.
[2] Huber, B. and Sturmfels, B. (1995), “A polyhedral method for solving sparse polynomial systems”, Math. Comp., 64(212), 1541–1555.
[3] Robert J. Walker (1950), Algebraic curves, New York: Dover Publications Inc.