## Weak Nullstellensatz for C

20 August, 2012

Weak Nullstellensatz says, that for an algebraically closed field ${\mathbb{K}}$ each maximal ideal ${I}$ in ${R=\mathbb{K}[X_1,\dots,X_n]}$ has form ${I=(X_1-a_1,\dots,X_n-a_n)}$, for ${a:=(a_1,\dots,a_n)\in \mathbb{K}^n}$, i.e. ${I=I(\{a\})}$, the ideal of an one-element subset of ${\mathbb{K}^n}$. Its proof in this generality needs quite a bit of commutative algebra. However, if we futher assume that ${\mathbb{K}}$ is uncountable (thus covering a very important case ${\mathbb{K}=\mathbb{C}}$) we can give a much quicker proof.

Theorem 1 Let ${\mathbb{K}}$ be an uncountable algebraically closed field, e.g. ${\mathbb{K}=\mathbb{C}}$, and ${I}$ a proper maximal ideal in ${R=\mathbb{K}[X_1,\dots,X_n]}$. Then there exists ${a:=(a_1,\dots,a_n)\in \mathbb{K}^n}$ such that ${I=I(\{a\})}$.

Proof: The first step is to show that ${R/I\cong\mathbb{K}}$. To see this, we will show that every ${b\in R/I}$ is algebraic, i.e. a root a nonzero polynomial ${f\in\mathbb{K}[z]}$. Note that the dimension of ${R/I}$ as a vectorspace over ${\mathbb{K}}$ is at most countable, as ${R/I}$ is generated by the images ${\phi(X^J)}$ of the monomials ${X^J}$ under the ring homomorphism ${\phi:R\rightarrow R/I}$, and the exponents ${J}$ form a countable set. Thus for ${b\in R/I\setminus\mathbb{K}}$ the set

$\displaystyle \left\{\frac{1}{b-t}\mid t\in \mathbb{K}\right\}$

is linearly dependent, i.e. there exist ${\mu_1,\dots,\mu_m}$ such that ${ \sum_{i=1}^m\frac{\mu_i}{b-t_i}=0. }$ Thus

$\displaystyle 0=\sum_{i=1}^m\frac{\mu_i}{b-t_i}=\frac{P(b)}{\prod_i{(b-t_i)}},$

where ${P\in\mathbb{K}[z]}$ and ${P(b)=0}$. As ${\mathbb{K}}$ is algebraically closed, we have that ${P}$ is linear, i.e. ${R/I\cong\mathbb{K}}$.

Next, we observe that ${\phi}$ maps ${R}$ to ${\mathbb{K}}$, and set ${a_i:=\phi(X_i)}$, for ${1\leq i\leq n}$. As ${\phi(X_i-a_i)=0}$, we see that ${X_i-a_i\in I}$, for ${1\leq i\leq n}$. By maximality of ${I}$, we obtain ${I=(X_1-a_1,\dots,X_n-a_n)}$, as claimed. $\Box$

Here one can find reduction of the general case (not assuming non-countability of $\mathbb{K}$) to this one.

## Schonhardt polyhedron and IMS-2013/4 program

28 May, 2012

One cannot always triangulate a non-convex polyhedron using only its vertices, sometimes one need to add more of them. A simple example of this phenomenon is Schonhardt polyhedron. Here is a picture illustrating how one builds it that I drew for a forthcoming paper, using Tikz LaTeX package, which is awesome, but totally overwhelming.

It fact, it’s easy to see that the 6 vertices and 12 edges it has are not enough. Indeed, each pair of non-intersecting edges determines a simplex, but it’s easy to observe that any such selection will include one the forbidden pairs of vertices AC, A’B, or B’C’. (The LaTeX source of the picture is here).

The paper I mention is related to a topic of the program on inverse moment problems at IMS (NUS/Singapore) in late 2013-early 2014 which I co-organize.

## Sage (sagemath.org) at GSoC 2012

19 March, 2012

Sage (not the accounting software, but sagemath.org) is accepted to Google Summer of Code 2012, and I will be one of many mentors.

## Elsevier and Springer Sue University Library

21 February, 2012

The battle is heating up! Now Elsevier, Springer and a smaller third publisher are suing a major university in Switzerland, the Eidgenössische Technische Hochschule Zürich, or ETH Zürich for short. Why? Because this university's library is distributing copies of their journal articles at a lower cost than the publishers themselves.

Aren't university libraries supposed to make journal articles available? Over on Google+, Willie Wong explains:

This is just SO wrong, that I find it hard to believe. I cannot boycott Springer, at least not yet, given that I have a number of papers under submission in Springer journals, but OK, I'm going to boycott Elsevier.

## Experimental Sage-based Mathematics course

7 August, 2011

We are to embark upon teaching a 2nd year undergraduate course Experimental Mathematics, which will cover computer algebra basics, and refresh concepts from 1st year linear algebra, calculus, and combinatoris, with few more advanced things thrown in. The course will be based on Sage, with the actual software running on dedicated servers, and accessible via Sage web notebook interface (i.e. basically nothing but a web browser running on the student’s computer/laptop/ipad, etc).

Given the enrollment of about 120, this will be interesting…

PS. Most students did not appreciate the freedom given, and complained, complained, complained…

## An email from Sendai (Tohoku University)

13 March, 2011

I was very glad today to receive an email from Akihiro Munemasa (from Tohoku university at Sendai), my long-term collaborator and friend. He says he’s OK, but has no electricity/water/gas…

And, as we see, they have internet, at least some kind of service (email went via me.com, a web-mail service run by Apple).

Well, I can only wish a lot of strength to Akihiro, and all the people affected by this disaster!

PS: today (14th March) he also sent me photos of his office:

## 2010 in review – Happy New Year!

2 January, 2011

While it might be snowing on this page, Edith picked up a mango fallen from a tree outside our block of flats this afternoon…

The stats helper monkeys at WordPress.com mulled over how this blog did in 2010, and here’s a high level summary of its overall blog health:

The Blog-Health-o-Meter™ reads Fresher than ever.

## Crunchy numbers

A Boeing 747-400 passenger jet can hold 416 passengers. This blog was viewed about 5,700 times in 2010. That’s about 14 full 747s.

In 2010, there were 6 new posts, growing the total archive of this blog to 33 posts. There were 5 pictures uploaded, taking up a total of 28kb.

The busiest day of the year was May 11th with 79 views. The most popular post that day was Is C++ the worst 1st programming language for maths majors?.

## Where did they come from?

The top referring sites in 2010 were terrytao.wordpress.com, cameroncounts.wordpress.com, debian-news.net, edventure.ntu.edu.sg, and symomega.wordpress.com.

Some visitors came searching, mostly for funny pictures, funny, confused cat, total unimodularity, and funny pics.

## Attractions in 2010

These are the posts and pages that got the most views in 2010.

1

Is C++ the worst 1st programming language for maths majors? May 2010

2

Total unimodularity and networks October 2009
1 comment

3

Cosets and normal subgroups March 2009

4

Bipartite matchings via linear programming September 2009
1 comment

5

Bellman-Ford algorithm for shortest paths and potentials September 2009