AI 中文总结
该研究以“$(a+b)^2 = a^2 + b^2$”的误差度量为切入点，引入实对数典范阈值（RLCT）与奇点消解，计算相关数学错误的RLCT并说明计算机代数软件的应用。
AI 中文摘要
对于“$(a+b)^2 = a^2 + b^2$”这个问题，有两种可能的答案：要么“这完全错误”，要么“我假设你处于特征2的域中”。如果选择第一个答案，我们可以自问：“它实际上错到什么程度？”在这篇说明性文章中，我们引入并使用奇点消解与实对数典范阈值（RLCT）作为此类误差的度量。RLCT是一个与极小模型纲领相关的不变量，且最近被发现与贝叶斯统计学、机器学习存在关联。受高中数学典型错误的启发，我们针对几类例子计算该不变量，其中包括上述“新生之梦”带来的误差，以及初等统计学中均值计算错误产生的误差。在部分例子中，我们还说明了如何利用计算机代数软件解决该问题。
英文摘要
There are two possible answers to "$(a+b)^2 = a^2 + b^2$". Either "this is completely wrong" or "I assume, you are in characteristic $2$". If we go for the first answer, we could ask ourselves: "How wrong is it actually?" In this expository article, we introduce and use resolution of singularities and the real log canonical threshold (RLCT) as a measure for such errors. The RLCT is an invariant with connections to the minimal model program and, as recently discovered and investigated, to Bayesian statistics and machine learning. Inspired by typical mistakes from high school mathematics, we compute this invariant for several classes of examples, among them the error arising from the "freshman's dream" above and from a flawed computation of means in elementary statistics. In some examples, we highlight how a computer algebra software can be used in order to solve this problem.
