A New Attempt at the Elementary Representation of Homogeneous Symmetric Polynomials

齐次对称多项式初等表示的新尝试 苏剑林 · 2013-07-05 · Mathematics algebra symmetry polynomial

This is a short paper I wrote for this semester's Advanced Algebra course. Speaking of which, I actually quite like the assessment style where instead of a final exam, your grade is based on regular coursework plus writing a paper, report, or doing experiments — I feel this gives a more comprehensive picture of one's knowledge and skills (though, more importantly, it's also because I enjoy writing~~). Our Advanced Algebra course consists of two parts: the regular lectures and a seminar, each graded separately. The instructor was kind to us and let the seminar be assessed through a short paper instead of an exam. Yeah~~

What I wrote about is symmetric polynomials. Actually, this article had been brewing for half a semester already, ever since we first learned about the elementary representation of symmetric polynomials. The so-called elementary representation means expressing a multivariate symmetric polynomial as a combination of $\sigma_1,\sigma_2,\sigma_3,...$. Here

$$\begin{aligned}\sigma_1=x_1+x_2+...+x_n \\ \sigma_2=x_1 x_2+x_1 x_3+...+x_1 x_n+x_2 x_3+...+x_{n-1} x_n \\ ... \\ \sigma_n=x_1 x_2 ... x_n\end{aligned}$$

The textbook gives the method of undetermined coefficients, but having to solve a system of equations every single time was quite bothersome to me, so I looked into a direct expansion approach, and eventually arrived at two methods. Around the same time I happened to be learning about tensors and came across the "Einstein summation convention," and wanting to make full use of its power led to this article. In fact, I defined two custom operations — the "square bracket" and the "round bracket" — both of which are notational simplifications. The two methods complement each other in a certain sense, and I'm quite satisfied with the result, so I'd like to share it with everyone. I won't paste the full content here — please download the PDF file to read it. more

Abstract
The fundamental theorem of symmetric polynomials tells us that every symmetric polynomial can be expressed as a polynomial in the elementary symmetric polynomials. But this is purely a theoretical guarantee — the concrete techniques for carrying out such transformations still need to be worked out. The Advanced Algebra textbook presents two different methods. One derives the representation term by term based on the leading term, but this is rarely used in practice because of its high computational cost. The second method uses undetermined coefficients, which is more efficient and faster. However, it is not hard to see that this method still has two shortcomings:
(1) Its "speed" is only relative. It is indeed fast for computer programming, but for manual computation it is still quite restrictive, since it reduces the problem to a system of linear equations in several variables, and solving such systems by hand is not easy.
(2) The process of undetermined coefficients does not reflect the intrinsic character of symmetric polynomials — it buries the regularity and beauty that "symmetry" brings to a polynomial.
In summary, it is worthwhile to explore new approaches to the elementary representation of symmetric polynomials. This paper is an attempt to do exactly that; without loss of generality, we restrict our attention to homogeneous symmetric polynomials in n variables. Through this investigation, I have obtained two methods that can give the elementary representation of a symmetric polynomial relatively quickly. In a certain sense they complement each other, and I will introduce both below.

Download: A New Attempt at the Elementary Representation of Homogeneous Symmetric Polynomials.pdf