Ciro Santilli @cirosantilli 40

mathoverflow.net/questions/11540/what-are-the-most-attractive-turing-undecidable-problems-in-mathematics/11557#11557 contains a good overview of the decidability status of variants over rings other than the integers.

www.jstor.org/stable/1970438 says for prime modulo there is an algorithm.

Question for non-prime modulo: math.stackexchange.com/questions/4944623/are-diophantine-equations-decidable-in-modular-arithmetic

TODO wtf is a "totient"? Where else is that word used besides in this concept?

E.g. International Mathematical Olympiad, International Physics Olympiad, competitive programming, etc.

Knowledge olympiads are stupid.

The most triggering thing possible is when programming competitions don't release their benchmarks as open source software afterwards: at least like that they might help someone to solve their real world problems. Maybe.

Some irrelevant people highlight that knowledge Olympiads can have good effects, because they are "an opportunity to meet university teachers and their research organizations". Ciro's argument is just that there are much more efficient ways to achieve those goals.

As an alternative way to get into university, this is not 100% horrible however, e.g. the University of São Paulo accepted students from olympiads in 2019 and then again 2023: jornal.usp.br/institucional/usp-oferece-200-vagas-em-mais-de-100-cursos-de-graduacao-para-alunos-participantes-de-olimpiadas-do-conhecimento/?a

Resources to study for knowledge olympiads:

Unlike over non-commutative rings, polynomials do look like proper polynomials over commutative ring.

In particular, Hilbert's tenth problem is about polynomials over the integers, which is a commutative ring, and therefore brings mindshare to this definition.

In this section we collect results about algebraic equations over more "exotic" fields

Generates numpy/fft_plot.svg, the plot of $2\sin (t)+\cos (4t)$ with 25 points and its DFT.

The output was also uploaded to: commons.wikimedia.org/wiki/File:DFT_2sin(t)_%2B_sin(4t).svg and added to en.wikipedia.org/w/index.php?title=Discrete_Fourier_transform&oldid=1176616763 only to be later removed of course: Deletionism on Wikipedia.

A hard problem ha been found for it, and it was called the "antihydra":