# Martin Orr's Blog

## Sleeper train and Spring

Posted by Martin Orr on Friday, 30 April 2010 at 20:29

I had a week at home in Belfast last week. Since I resolved a couple of years ago not to fly for routine journeys, it is quite a long journey between Belfast and Paris. In the past I have always done this with a night's stop in England, taking most of two days. This time I broke my journey Paris to Belfast in Leeds for the Future Sounds of Swing weekend. This was almost the only lindy classes I have gone to since last June. It was a great weekend.

On my way back from Belfast to Paris, I took the sleeper train from Edinburgh to London. This allowed me to get from Belfast to Paris in 24 hours (with a bit more time required for travel at either end). It takes about 7 hours leaving at 11.40. The train during the day, which makes several stops, takes less than 5 hours. I suppose it goes slower to give a smoother ride and so that you can get a decent night's sleep. In fact you can stay in your cabin for another hour after it arrives. It cost me £35 with a rail card.

The room is very small with not much more than a pair of bunk beds and a sink. When I saw it I worried whether there would be space for my suitcase, but there was a shelf for it. I was sharing with one other man. It was very comfortable and the movement was barely noticeable. You get brought tea or coffee and a biscuit when the train arrives (in a disposable cup, I suppose so you can take it away if you want).

When I got back to Bures, as soon as I looked across the valley I could see that something had changed. The woods which cover all the upper part of the valley side had gone from brown to green. The leaves have come out on all the deciduous trees on the campus, which was only just starting when I left. This makes everything greener but less open.

Tags lindy, paris, train, travel

## Hensel's lemma and algebraic functions

Posted by Martin Orr on Monday, 05 April 2010 at 22:34

An algebraic function is a function which we obtain by solving a polynomial in two variables and to write as a function of . In general polynomials have more than one root, so (informally) we get a multi-valued function. In this post I will restrict attention to regions of the plane in which we can unambiguously pick a single "branch" of the function. What happens where branches meet will be the subject of a later post.

I shall give an algorithm for expressing an algebraic function (in such a nicely behaved region) as a power series, thereby proving that a power series solution to the original polynomial exists. The generalisation of this result to a complete discrete valuation ring is Hensel's lemma, and is particularly important to number theorists in the case of p-adic integers (which were invented by Hensel). In this post I will focus on the case of algebraic functions, as it is easier to apply geometric intuition.