Samstag, Oktober 03, 2009

Ehrhart polynomials and integer points in polytopes

It's about time for me to write about something mathematical on this blog. I used the opportunity to experiment with MathML and SVG. Unfortunately, the state of these technologies is rather horrible, which is why I can't write the actual entry in the blog itself. HTML 5 promises to improve things, but it's not quite there yet.

So here is a link to my text on Ehrhart polynomials.

There's a little bit of backstory here which I should probably mention. I was reading up on Ehrhart polynomials a while ago, and in particular I was looking for a proof of their existence. Unfortunately, the proofs I found immediately by perusing literature used rather abgewandte Mathematik, which made me sad. So, in a moment of the kind of hubris which is necessary to do these kinds of things, I decided that I could find an elementary proof on my own. I succeeded, and I thought to myself, "Hey, that proof is actually rather simple. I've been looking for something mathematical to write up on my blog, let's just use this."

So I started, and I had this goal in mind that I could explain my proof in a way that is understandable to ordinary laypeople. In the process, I had to admit to myself that the proof is probably not that simple.

You see, I am not writing for the kind of people who are uninterested in mathematics - that would be futile - but I do want my writing to be interesting and useful for other students of mathematics and interested laypeople. Sometimes, I like to try to write a text where my yardstick is, "Would I have been able to follow and appreciate this text at the beginning of my university studies?" Of course it is not always feasible to write texts like that, and it is actually incredibly hard to tell whether I achieve this goal because I have mostly forgotten who I was five years ago. Trying to see things from that older perspective is not easy.

I do hope that I have succeeded, and while the MathML was annoying to write, it was ultimately enjoyable because I could touch a large number of ideas and areas that are relevant to my daily work.

In the future, I will probably experiment with ASCIIMathML, which I discovered a bit too late. It appears to offer a reasonable solution to the verbosity of MathML.

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