Writing polynomials in terms of invariants and coinvariants

There is a fun little fact regarding polynomials in two variables $x$ and $y$:

Any two-variable polynomial $f(x,y)$ can be uniquely written as a sum of a symmetric polynomial and an antisymmetric polynomial.

(To be more precise, this is true for polynomials over any field of characteristic not equal to $2$. For simplicity, in what follows we will assume that our polynomials have coefficients in $\mathbb{C}$.)

Recall that a polynomial $g$ is symmetric if it does not change upon permuting its variables. In this case, with two variables, $g(x,y)=g(y,x)$. It is antisymmetric if swapping any two of the variables negates it, in this case $g(x,y)=-g(y,x)$.

It is not hard to prove the fact above. To show existence of the decomposition, set $g(x,y)=\frac{f(x,y)+f(y,x)}{2}$ and $h(x,y)=\frac{f(x,y)-f(y,x)}{2}$. Then \[f(x,y)=g(x,y)+h(x,y),\] and $g$ is symmetric while $h$ is antisymmetric. For instance, if $f(x,y)=x^2$, then we can write \[x^2=\frac{x^2+y^2}{2}+\frac{x^2-y^2}{2}.\]

Expii: Learning, connected

It’s been several months since I posted a gemstone, and the main reason is that much of my free-time mathematics energy recently became channeled into a new project: Expii.

Expii (currently beta) is a new online crowdsourced learning site that aims to fill the gaps in users’ understanding of topics, with the goal of making math, science, and other topics easy for everyone in the universe. Its motto? Learning, connected.

With an addictive, game-like format (hence the XP pun) in which users are awarded “fame points” for writing good explanations and “experience points” for successfully making it through tutorials, Expii is more interactive and community oriented than other online learning resources like Wikipedia. It is also more structured than question-and-answer sites like Quora or Stack Exchange, in that the primary “graph structure” for the topics is organized by our team, and users fill in the content in the nodes.

PEMDAS is broken (and how we can fix it)

In the first week of teaching my Calculus 1 discussion section this term, I decided to give the students a Precalc Review Worksheet. Its purpose was to refresh their memories of the basics of arithmetic, algebra, and trigonometry, and see what they had remembered from high school.

Surprisingly, it was the arithmetic part that they had the most trouble with. Not things like multiplication and long division of large numbers - those things are taught well in our grade schools - but when they encountered a complicated multi-step arithmetic problem such as the first problem on the worksheet, they were stumped:

Simplify: $1+2-3\cdot 4/5+4/3\cdot 2-1$

Gradually, some of the groups began to solve the problem. But some claimed it was $-16/15$, others guessed that it was $34/15$, and yet others insisted that it was $-46/15$. Who was correct? And why were they all getting different answers despite carefully checking over their work?

A proof of the existence of finite fields of every possible order

This is our first contributed gemstone! Submitted by user Anon1.

In the following, $p$ denotes a prime. We wish to prove that, for all positive integers $n$, there is a finite field of order $p^{n}$. Step 1. Restating the problem.

Claim: It suffices to show that, for some power of $p$ (call it $q$), there exists a finite field of order $q^{n}$.

Proof. Suppose there is a field $F$ such that $|F| = q^{n}$. The claim is that the solution set to $x^{p^{n}} = x$ in $F$ is a subfield of order $p^{n}$.

Since $q$ is a power of $p$, we have \[p^{n}-1 | q^{n}-1.\] Since $q^{n}-1$ is the order of the cyclic group $F^{ \times }$, we know that $F^{ \times }$ has a unique cyclic subgroup of order $p^{n}-1$. This contributes $p^{n}-1$ solutions to $x^{p^{n}} = x$ in $F$. But $0$ is another solution, since it is not an element of $F^{ \times }$. This gives $p^{n}$ solutions, so this gives all solutions.

FindStat and combinatorial statistics

Last semester, I attended Sage Days 54 at UC Davis. In addition to learning about Sage development (perhaps a topic for a later blog post), I was introduced to FindStat, a new online database of combinatorial statistics.

You may be familiar with the Online Encyclopedia of Integer Sequences; the idea of FindStat is similar, and somewhat more general. The Online Encyclopedia of Integer Sequences is a database of mathematically significant sequences, and to search the database you can simply enter a list of numbers. It will return all the sequences containing your list as a consecutive subsequence, along with the mathematical significance of each such sequence and any other relevant information.

FindStat does the same thing, but with combinatorial statistics instead of sequences. A combinatorial statistic is any integer-valued function defined on a set of combinatorial objects (such as graphs, permutations, posets, and so on). Some common examples of combinatorial statistics are:

• The number of edges of a finite simple graph,
• The length of a permutation, that is, the smallest length of a decomposition of the permutation into transpositions,
• The number of parts of a partition,
• The diameter of a tree.