This is a short paper I wrote for my Higher Algebra course this semester. Speaking of which, I actually quite like the grading methods that do not involve exams but instead evaluate performance through regular assessments, writing papers, reports, or conducting experiments. After all, I feel this is a more comprehensive reflection of one’s level of knowledge and skills (and of course, a more important reason is that I simply enjoy writing). Our Higher Algebra curriculum consists of two parts: the basic lectures and the seminar course, which are assessed separately. Our teacher was very considerate; the seminar course required no exam, only a short paper. Yeah!
I chose to write about symmetric polynomials. In fact, this article had been brewing for half a semester, starting from when we first learned about the elementary representation of symmetric polynomials. The so-called elementary representation involves expressing a multivariate symmetric polynomial as a combination of \sigma_1, \sigma_2, \sigma_3, \dots, where:
\begin{aligned} \sigma_1 &= x_1 + x_2 + \dots + x_n \\ \sigma_2 &= x_1 x_2 + x_1 x_3 + \dots + x_1 x_n + x_2 x_3 + \dots + x_{n-1} x_n \\ &\dots \\ \sigma_n &= x_1 x_2 \dots x_n \end{aligned}
The textbook provides the method of undetermined coefficients, but it requires solving a system of equations every time, which I found quite bothersome. Therefore, I researched a scheme for direct expansion and eventually derived two methods. At that time, I was also just coming into contact with tensor theory and learned about the "Einstein summation convention." I wanted to fully utilize its power, which led to the creation of this article. In fact, I defined two types of operations, "square brackets" and "round brackets," which are both symbolic simplifications. The two methods complement each other in a sense, and I am quite satisfied with the result, so I am sharing it with everyone. I won’t post the specific details here; please download the PDF file to read it.
Abstract
The fundamental theorem of symmetric polynomials tells us that every symmetric polynomial can be expressed as a polynomial of elementary symmetric polynomials. However, this is merely theoretical; specific transformation techniques remain to be explored. The "Higher Algebra" tutorial provides two different methods. One is to obtain the representation successively based on the leading term, but this is not frequently used due to its high computational cost. The second method is the method of undetermined coefficients, which is more efficient and faster. However, it is not difficult to find that it still has the following two shortcomings:
(1) Its "speed" is relative. While it is indeed fast for computer programming, it is still quite limited for manual calculation. After all, it converts the problem into a system of linear equations in multiple variables, and solving such systems manually is not easy.
(2) The process of the method of undetermined coefficients does not reflect the characteristics of symmetric polynomials, obscuring the patterns and aesthetic beauty of "symmetry" within polynomials.
In summary, it is necessary to conduct new explorations into the elementary representation of symmetric polynomials. This paper attempts such an exploration. Without loss of generality, we only consider n-variable homogeneous symmetric polynomials. Through research, the author has obtained two methods that can relatively quickly provide the elementary representation of symmetric polynomials. In a sense, they are complementary to each other, and the author will introduce them below.
Download: A New Attempt at the Elementary Representation of Homogeneous Symmetric Polynomials.pdf
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