我們證明：若 a 或 b 至少有一者不屬於常數域 F，則方程僅在 N = 1 時有解。當 a、b ∈ F 時，N 可以取為常數域 F 的基數之任意冪次。此外，若方程對某個 F 的特徵 p 的冪次 N 有解，則對所有更大的 p 的冪次亦有解，因此該遞迴數列在此類指標上具有無限多個零點。另一方面，若 p 不整除 N，則當 K 的虧格 g_K = 0 時有 N ≤ 2；當 g_K = 1 時有 N ≤ 3。
最後，我們提出一個演算法，用以判定此類遞迴數列是否具有有限或無限多個零點。;In this thesis， we study the zero multiplicities of simple ternary linear recurrence sequences over a global function field K of genus 0 or 1， assuming that the second term of the sequence is zero. The problem reduces to determining integers N for which the Diophantine equation aX^N + bY^N = 1 admits a nonzero solution (X， Y) = (u， v) in K^2， where a and b are fixed nonzero elements of K.
We prove that if at least one of a or b does not lie in the constant field F， then the equation admits only the solution N = 1. When a and b belong to F， N can be any power of the cardinality of F. Moreover， if the equation admits a solution for some power of the characteristic p of F， then it also admits solutions for all larger powers of p. Consequently， the recurrence sequence has infinitely many zeros at such indices. On the other hand， if p does not divide N， then N <= 2 when the genus g_K = 0， and N <= 3 when g_K = 1.
Finally， we present an algorithm to determine whether such a recurrence sequence has finitely or infinitely many zeros.
