Symmetry and Algebraic Equations

从对称角度看代数方程 苏剑林 · 2011-04-29 · Mathematics equation algebra symmetry group theory

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These days, BoJone has become fascinated by two things: the principle of least action and symmetry. Both of these occupy an almost central place in physics. As discussed before, the principle of least action allows us to construct the entire framework of modern physics, reflecting nature's "economical mindset." Symmetry, on the other hand, is the embodiment of conservation, and also corresponds to nature's "sense of beauty." This post focuses on symmetry at its simplest level.

Symmetric things matter, and they are beautiful. Here, of course, I mean symmetry in the mathematical sense. Many problems in mathematics can be written in a symmetric form, and precisely because of that symmetry, the process of solving them is generally much simpler than for asymmetric expressions. It's said that the most cutting-edge frameworks in modern physics (including general relativity) are described using group theory, and group theory is exactly the powerful tool used to study symmetry. This shows that symmetry, and symmetry-based methods, have extensive applications in practice. (Of course, this post won't discuss group theory — the main reason being that BoJone doesn't actually understand group theory... ^_^)

Let's start with the quadratic equation. By Vieta's formulas, any quadratic equation can be written as:

$$\begin{aligned}x_1+x_2=a \\ x_1 x_2=b\end{aligned}$$

This is a highly symmetric form! The symmetry here shows up in the fact that swapping $x_1,x_2$ with each other leaves the equation unchanged. If we set $x_1=y_1+y_2,x_2=y_1-y_2$, this becomes

$$2y_1=a,y_1^2-y_2^2=b$$

which quickly gives us $y_1,y_2$, and from there we can find both roots of the equation. more

BoJone originally hoped to push this method further and apply it to the cubic equation, but unfortunately, given limited mathematical ability, that task couldn't be completed. However, given the known formula for the roots of the cubic, this method can be extended to the quartic equation. Any quartic equation can be written as

$$\begin{aligned}x_1+x_2+x_3+x_4=a \\ x_1 x_2+x_1 x_3+x_1 x_4+x_2 x_3+x_2 x_4+x_3 x_4=b \\ x_1 x_2 x_3+x_1 x_2 x_4+x_1 x_3 x_4+x_2 x_3 x_4=c \\ x_1 x_2 x_3 x_4=d\end{aligned}$$

Setting $x_1=y_1+y_2,x_2=y_1-y_2,x_3=y_3+y_4,x_4=y_3-y_4$, the system becomes

$$\begin{aligned}2(y_1+y_3)=a \\ (y_1^2-y_2^2)+(y_3^2-y_4^2)+4x_1 x_3=b \\ 2(y_1^2-y_2^2)y_3+2(y_3^2-y_4^2)y_1=c \\ (y_1^2-y_2^2)(y_3^2-y_4^2)=d\end{aligned}$$

If we then let $z_1=y_1^2-y_1^2,z_3=y_3^2-y_4^2$, another symmetric system of equations emerges:

$$\begin{aligned}2(y_1+y_3)=a \\ z_1+z_3+4y_1 y_3=b \\ 2z_1 y_3+2z_3 y_1=c \\ z_1 z_3=d\end{aligned}$$

Continuing with a similar substitution $y_1=w_1+w_2,y_3=w_1-w_2,z_1=w_3+w_4,z_3=w_3-w_4$, we get

$$\begin{aligned}4w_1=a \\ 2w_3+4(w_1^2-w_2^2)=b \\ 4(w_1 w_3-w_2 w_4)=c \\ w_3^2-w_4^2=d\end{aligned}$$

At this point, the symmetry is essentially exhausted, so let's try solving it with a more conventional method (substitution and elimination). Since $w_1$ can be found directly, it's a known quantity. Multiplying the second equation by $2w_1$ and subtracting the third equation gives

$$\begin{aligned}8w_1 (w_1^2-w_2^2)+4w_2 w_4=2w_1 b-c \\ w_4=\frac{2w_1 b-c-8w_1^3}{4w_2}+2w_1 w_2\end{aligned}$$

and from the second equation we also have $w_3=\frac{b}{2}-2(w_1^2-w_2^2)$

Substituting $w_3,w_4$ into $w_3^2-w_4^2=d$ then gives us a cubic equation in $w_2^2$, which can be solved. Done!

Summary

Symmetry is, in fact, a topic of remarkable depth — its ubiquity and the breadth of its applications never cease to amaze us. Even in the aesthetics of everyday life, we tend to gravitate toward objects that possess symmetry. Likewise, in research across mathematics, physics, chemistry, and other fields, transforming a problem into a symmetric form generally makes things easier, and repeated symmetric transformations gradually simplify the problem at hand. Mastering the methods for studying symmetry is therefore extremely important. Of course, strictly speaking, the examples above hardly qualify as genuine symmetry analysis — they're really just some off-the-cuff reflections BoJone had while poking around. The most powerful tool for this purpose today is still group theory. While reading Differential Equations and Problems in Mathematical Physics, I came across the author's remark that Lie group analysis is the most effective method for solving differential equations, which greatly sparked my interest in group theory. And in the course of studying Theoretical Mechanics, I likewise discovered just how widely group theory is applied. So BoJone has come to a clear realization: it's time to start learning group theory...