数域上牛顿法的收敛点个数
On the Number of Places of Convergence for Newton's Method over Number Fields
摘要：
设f是数域K中系数至少为2次的多项式，x_0是K的充分一般元素，α是f的根.本文给出了从点x_0开始的Newton迭代法对K的无穷多个位置v v-渐近收敛于根α的精确条件。作为一个推论，我们表明，如果f是不可约的K度至少为3，那么牛顿迭代收敛v-adically任何给定的根f的无穷多个地方v。我们还推测，一组的地方牛顿迭代发散有全密度，并给出一些启发式和数值证据。
英文摘要：
Let f be a polynomial of degree at least 2 with coefficients in a number field K, let x_0 be a sufficiently general element of K, and let alpha be a root of f. We give precise conditions under which Newton iteration, started at the point x_0, converges v-adically to the root alpha for infinitely many places v of K. As a corollary we show that if f is irreducible over K of degree at least 3, then Newton iteration converges v-adically to any given root of f for infinitely many places v. We also conjecture that the set of places for which Newton iteration diverges has full density and give some heuristic and numerical evidence.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
