# Blog chronicle for all time

On prose in math papers Sat, 28 May 2022

This is an apology for writing prose in mathematics papers (and a clarification of what I mean by “prose”).

Some properties of varieties over the complex numbers do not change when the variety (via its defining equations which must have integral coefficients) is instead considered over sufficiently large finite fields of sufficiently large characteristic. This phenomenon can be explained via symbolic algorithms in computer algebra, particularly Gröbner bases.

I describe the two more subtle problems I used to run into when submitting papers to the arXiv and their solutions.

I present a Nugs.net downloader I wrote and which is now usable.

Any open cover of a primary basic semialgebraic set can be refined to a locally finite one. Here is an explicit proof of this fact when the cover consists of neighborhoods indexed by points from the set itself and which, after an initial adjustment of the sizes of the open sets, constructs a locally finite subcover, i.e., just removes redundant open sets. This detail matters when the points indexing the original cover have special significance elsewhere.

I give a counterexample to a claim in the book “Realization spaces of polytopes”. This counterexample shows that a fundamental definition in this book is (slightly, technically) wrong. I explain what causes the failure and how it might be fixed in a forthcoming note.

On experimental mathematics Sat, 17 Jul 2021

Experimental mathematics uses computers to inspire research in terms of finding examples and counterexamples, suggesting conjectures or informing proof strategies. I present a strange phenomenon in this practice at two examples: it happens that mathematical software confirms conjectured properties of very large objects very quickly. Much, much more quickly than one might expect a general problem of that type to be solved. In these cases, the software apparently knows a “proof” of the conjecture and the human can either look it up in the source code or at least be more confident in the conjecture.

In his article “Gaussian Representation of Independence Models over Four Random Variables” Petr Šimeček gives realizations of all conditional independence models arising from (not necessarily regular) Gaussian distributions in dimension 4. Only one of these models, M85, is given with irrational correlations. Here I give a rational representation of this model. The question whether all Gaussian CI models are realizable by rational correlation matrices is still open and M85 was the only concrete undecided example I am aware of.

tappp.hpp v0.2.0 released Thu, 27 Feb 2020

tappp.hpp is a header-only TAP producer for C++17.

The symmetry reduction of LUBF-gaussoids in dimension 8 was a tough nut. I learned that precomputing invariants of the symmetry classes first allows more intelligent distribution of tasks in a parallel symmetry reduction algorithm. In this case, I achieved a 160-fold speedup.

Yet another static site generator – for a blog and homepage.