Equatorial Mathematics

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.

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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.

Tags: algebra, combinatorics
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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.

Tags: maths, python, software, teaching
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Elsevier and Springer Sue University Library

21 February, 2012

Reblogged from Azimuth:

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:

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…

Tags: MTH213, python, software, teaching
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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:

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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…

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Equatorial Mathematics

Weak Nullstellensatz for C

Schonhardt polyhedron and IMS-2013/4 program

Sage (sagemath.org) at GSoC 2012

Elsevier and Springer Sue University Library

Experimental Sage-based Mathematics course

An email from Sendai (Tohoku University)

2010 in review – Happy New Year!

Crunchy numbers

Where did they come from?

Attractions in 2010

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