有理数的HTTP-完全环

摘要：

对于环$R$，希尔伯特第十问题$HTP(R)$是在$R$上的多项式方程组，在多个变量中，解在$R$中。我们将$HTP$视为枚举运算符，将每个素数集合$W$映射到$HTP(\mathbb Z[W^{-1}])$，后者自然被视为$\mathbb Z[X_1,X_2,\ldots]$中的一组多项式。已知对于几乎所有的$W$，跳跃$W'$不$1$ -减少到$HTP(R_W)$。相反，我们证明了每一个图灵度都包含一个$W$集合，对于这个集合，这样的$1$ -归约确实成立：这些$W$被称为“HTTP-完全的”。“继续，我们得出额外的结果，关于不可能的决策程序$W'$从$HTP(\mathbb Z[W^{-1}])$可以成功一致的一组措施$1$，以及关于后果的边界集的$HTP$运营商的情况下，$\mathbb Z$有一个存在的定义在$\mathbb Q$。

英文摘要：

For a ring $R$, Hilbert's Tenth Problem $HTP(R)$ is the set of polynomial equations over $R$, in several variables, with solutions in $R$. We view $HTP$ as an enumeration operator, mapping each set $W$ of prime numbers to $HTP(\mathbb Z[W^{-1}])$, which is naturally viewed as a set of polynomials in $\mathbb Z[X_1,X_2,\ldots]$. It is known that for almost all $W$, the jump $W'$ does not $1$-reduce to $HTP(R_W)$. In contrast, we show that every Turing degree contains a set $W$ for which such a $1$-reduction does hold: these $W$ are said to be "HTP-complete." Continuing, we derive additional results regarding the impossibility that a decision procedure for $W'$ from $HTP(\mathbb Z[W^{-1}])$ can succeed uniformly on a set of measure $1$, and regarding the consequences for the boundary sets of the $HTP$ operator in case $\mathbb Z$ has an existential definition in $\mathbb Q$.

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