# Shortform Posts

A section of the website for shortform posts, still technical, but separate from the main blog and with a lower threshold for quality.

- Barycentric Lagrange Interpolation

In my studies of the Remez algorithm, I learned about the barycentric Lagrange interpolation formula. The context is finding a polynomial of degree at most $n$ that passes through $n+1$ points $(x_0, y_0), \dots, (x_n, y_n)$. The classical Lagrange interpolation formula is what you’d write down if you “just did it.” $$f(x) = \sum_{i=0}^n y_i \cdot \prod_{j \neq i}\frac{x - x_j}{x_i - x_j}$$ I wrote a 2014 article deriving this more gently, and implementing it in Haskell for secret sharing. - Random life updates

I added Twitter syndication, and because I have nothing to test it with I’ll share some random life updates. My daughter was born recently, which means I’m on paternity leave for a few months. Hopefully in the liminal hours of sleep training, I’ll have some time to work on my book. Or at least catch up on reading. I finally published my grandmother’s autobiography. I doubt anyone outside of my family will want to read it, but it’s available for sale on Amazon. - Math databases

Steven Clontz informed me of an effort he’s involved in called code4math. It’s described as a professional organization for the advancement of mathematical research through building non-research software infrastructure. By that he means, for example, writing software packages like Macaulay2 or databases of mathematical objects that other researchers can use to do their research. Clontz recently gave a talk on the topic, with ample discussion of the evaluation material they can provide to justify the academic value of this sort of work. - Experiments with POSSE

POSSE stands for Publish (on your) Own Site, Syndicate Elsewhere. I first heard about it from Cory Doctorow. I’m experimenting with automation to convert posts tagged shortform into Mastodon threads (I’m mathstodon.xyz/@j2kun). I’m using Hugo as a static site generator, with the source a (private) GitHub repository, and Netlify for deployments. After a deployment, Netlify calls a serverless function that hits the GitHub API with a POST request to trigger a GitHub action workflow. - Remez and function approximations

I’ve been learning recently about how to approximate functions by low-degree polynomials. This is useful in fully homomorphic encryption (FHE) in the context of “arithmetic FHE” (see my FHE overview article), where the computational model makes low-degree polynomials cheap to evaluate and non-polynomial functions expensive or impossible. In browsing the state of the art I came across two interesting things. The first is the software package lolremez that implements polynomial (and rational polynomial $f(x) / g(x)$) function approximation using the so-called Remez algorithm.