有限交换环上随机线性方程的可满足性阈值

中文摘要 AI 辅助

我们将有限域上随机线性方程的研究扩展到有限交换环上的方程。我们精确刻画了可满足性阈值在亚线性尺度上出现的情形；即当约束数量$m$相对于变量数量$n$为亚线性时，随机系统以高概率变得不可满足。在这一范围内，我们确定了可满足性阈值的精确值。在可满足性阈值为$n$的线性函数的互补范围内，当$R$为主环时，我们确定了其精确值。有趣的是，该值与$R$的选择无关，这与$R$为有限域时的现象相同。我们进一步证明，如果$R$非主环，这种对$R$的独立性会被打破。特别地，我们研究了一个经典的非主环族，并确定了该族中所有环的可满足性阈值。值得注意的是，在这种情况下，可满足性阈值不仅依赖于底层环，还依赖于定义随机线性方程模型的其他参数。

英文摘要

We extend the study of random linear equations over finite fields to equations over finite commutative rings. We characterize precisely when the satisfiability threshold occurs at a sublinear scale; namely, when the random system become unsatisfiable with high probability with a number of constraints $m$ that is sublinear in $n$, the number of variables. In this regime, we determine the exact value of the satisfiability threshold. In the complementary regime where the satisfiability threshold is linear in $n$, we determine its precise value when $R$ is a principal ring. Interestingly, this value is independent of the choice of $R$, mirroring the same phenomenon when $R$ is a finite field. We further prove that this independence of $R$ breaks down if $R$ is nonprincipal. In particular, we investigate a classical family of nonprincipal rings and determine the satisfiability thresholds for all rings in this family. Remarkably, in this setting, the satisfiability threshold depends not only on the underlying ring, but also on other parameters defining the random linear equation model.

发表机构

• University of Waterloo（滑铁卢大学）

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