Abstract

. We show that a plane algebraic curve f = 0 over the complex numbers has on it either at most 22V (f) points whose coordinates are both roots of unity, or infinitely many such points. Here V (f) is the area of the Newton polytope of f. We present an algorithm for finding all these points. 1. Introduction. When does a curve f(x, y) = 0 with complex coe#cients have cyclotomic points on it? By cyclotomic points we mean points (x, y) with x and y both roots of unity. How do we estimate the number of cyclotomic points on a given curve? How do we actually find all these points? In Section 2 we present an algorithm for finding the cyclotomic part of a polynomial in one variable. We do this first for polynomials with rational coe#cients, and then show how the algorithm can be extended to polynomials with complex coe#cients. In Section 3 we give an algorithm for finding all the cyclotomic points on a curve. This algorithm uses the algorithm of Section 2. Again, we first present an algorithm ...

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