Algebraic Equations from the Perspective of Symmetry

Lately, BoJone has become fascinated by two things: the Principle of Least Action and Symmetry. These two "concepts" hold a position of paramount importance in physics. As mentioned previously, the entire framework of modern physics can be constructed through the Principle of Least Action, reflecting the "economic mind" of nature. The latter, symmetry, is the manifestation of conservation and corresponds to the "aesthetic sense" of the natural world. This article aims to discuss symmetry from a very fundamental level.

Symmetrical things are important and beautiful. Of course, I am referring to mathematical symmetry. Many problems in mathematics can be formulated into symmetrical expressions, and due to this symmetry, the solving process is generally much simpler than that of asymmetrical expressions. It is said that the most cutting-edge frameworks in contemporary physics are described using Group Theory (including General Relativity), and Group Theory is precisely a powerful tool for studying symmetry. (Of course, this article will not discuss Group Theory, primarily because BoJone does not understand it yet... ^_^)

Let us first look at quadratic equations. According to Vieta’s formulas, any quadratic equation can be expressed in the following form: \begin{aligned} x_1 + x_2 &= a \\ x_1 x_2 &= b \end{aligned}

This is a highly symmetrical form! The symmetry here is reflected in the fact that the form of the equations remains unchanged if x_1 and x_2 are swapped. If we let x_1 = y_1 + y_2 and x_2 = y_1 - y_2, the system becomes: 2y_1 = a, \quad y_1^2 - y_2^2 = b

In this way, y_1 and y_2 can be found very quickly, and subsequently, the two roots of the equation can be determined.

BoJone had hoped to follow this logic and extend this method to cubic equations, but unfortunately, his mathematical level is limited, and he could not complete that task. However, given the existence of the cubic formula, this method can be generalized to quartic equations. Any quartic equation can be represented 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}

Let 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 of equations transforms into: \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 let z_1 = y_1^2 - y_1^2 and z_3 = y_3^2 - y_4^2, another symmetrical system of equations appears: \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 obtain: \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 no longer apparent, so we attempt to solve using conventional methods (substitution and elimination). Since w_1 can be solved directly, it is a known quantity. Multiplying the second equation by 2w_1 and subtracting the third equation, we get: \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}

Simultaneously, from the second equation, we have: w_3 = \frac{b}{2} - 2(w_1^2 - w_2^2)

Substituting w_3 and w_4 into w_3^2 - w_4^2 = d yields a cubic equation in terms of w_2^2, which can be solved. Done!