Estimation of calculation errors and resolution of singularities

Abstract

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.