A Finiteness Result for Post-critically Finite Polynomials

作者：Ingram Patrick^*

来源：International Mathematics Research Notices, 2012, (3): 524-543.

DOI：10.1093/imrn/rnr030

Let P-d denote the moduli space of polynomials of degree d, up to affine conjugacy. We show that the set of points in P-d(C) corresponding to post-critically finite polynomials is a set of algebraic points of bounded height. It follows that for any B, the set of conjugacy classes of post-critically finite polynomials of degree d with coefficients of algebraic degree at most B is a finite and effectively computable set. As an example, we exhibit a complete list of representatives of the conjugacy classes of monic post-critically finite cubic polynomials in Q[z]. The proof of the main result comes down to finding a relation between the natural height on P-d, and Silverman%26apos;s critical height.

• 出版日期2012