AI 中文总结
本文针对二次域整数环$O_K$，证明其16未知量多项式方程求解问题不可判定，实二次域时仅需15个未知量，完善了希尔伯特第十问题相关不可判定性结论。
AI 中文摘要
设$K$为任意二次数域，$O_K$为$K$中的代数整数环。1975年J. Denef证明了$O_K$上的希尔伯特第十问题无解。本文确立如下不可判定性结果：不存在算法判定任意给定的16个未知量、整数系数的多项式方程$P(z_1,\ldots,z_{16})=0$是否在$O_K$上有解；当$K$为实二次域时，15个未知量即可满足不可判定性要求。
英文摘要
Let $K$ be any quadratic number field, and let $O_K$ be the ring of algebraic integers in $K$. In 1975 J. Denef proved that Hilbert's Tenth Problem over $O_K$ has a negative solution. In this paper we establish the following undecidability result: There is no algorithm to decide whether an arbitrarily given polynomial equation $P(z_1,\ldots,z_{16})=0$ (with integer coefficients and 16 unknowns) has solutions over $O_K$. Moreover, when $K$ is a real quadratic field, we show that $15$ unknowns suffice for undecidability.
Comments11 pages
