In this article, we will revisit the topic of symmetry, but this time the subject is not "equalities" but "inequalities."
In mathematical research, we often encounter various functional expressions, a significant portion of which are "symmetric." What is a symmetric function? There are many ways to describe symmetry, but for multivariate symmetric expressions, our definition is a function that satisfies f(x_1, x_2, \dots, x_n) = f(y_1, y_2, \dots, y_n), where (y_1, y_2, \dots, y_n) is any permutation of (x_1, x_2, \dots, x_n). In common terms, if you swap any two unknowns in the expression, the resulting expression remains identical to the original. For example, \sin x + \sin y becomes \sin y + \sin x after swapping x and y, which is the same as the original. Similarly, for xy + yz + zx, swapping y and z gives xz + zy + yx, which is also the same. If a symmetric function is a polynomial of degree n, it is called an n-th degree symmetric polynomial. The example xz + zy + yx is a ternary quadratic symmetric polynomial.
What are the benefits of symmetry?
I have mentioned in previous articles that symmetry helps us in finding solutions. For instance, it is easier to find first integrals in symmetric physical systems. Noether’s Theorem also tells us that the symmetry of a physical system corresponds to conservation laws. Of course, these statements are quite abstract and require knowledge of theoretical physics to be deeply appreciated. So, for us, how can symmetry help? Let’s look at one example: polynomial expansion.
Suppose we want to expand (a+b+c)^3. Besides calculating it step-by-step according to arithmetic rules, we have another method: the method of undetermined coefficients. This is also the method used in computer algebraic proofs. We know that the expansion of (a+b+c)^3 must contain the terms a^3, b^3, c^3. Since (a+b+c)^3 is a symmetric expression—in other words, the "roles" of a, b, c are equivalent—the expansion must contain a part in the form \lambda_1 (a^3+b^3+c^3). Next, we can reason that the expansion must contain the term a^2b, and consequently the five terms a^2c, b^2a, b^2c, c^2a, c^2b. We should not focus on the specific letters a, b, c, but rather see their essence: they are all the square of one unknown multiplied by another unknown. Thus, the roles of these six terms are also equivalent, and the expansion must contain \lambda_2 (a^2b + a^2c + b^2a + b^2c + c^2a + c^2b). The final term is \lambda_3 abc. Therefore, we know:
\begin{aligned} &(a+b+c)^3 \\ =&\, \lambda_1 (a^3+b^3+c^3) + \lambda_2 (a^2b + a^2c + b^2a + b^2c + c^2a + c^2b) + \lambda_3 abc \end{aligned}
To find the values of \lambda_1, \lambda_2, \lambda_3, we only need to substitute three sets of data to obtain the specific values. This is a system of three linear equations. Imagine if a, b, c were not symmetric; we would have to assign an unknown coefficient to every single term (10 terms in total), substitute data, and solve a system of ten linear equations. Clearly, symmetry significantly reduces the amount of calculation. In fact, there are many more benefits to symmetry; shall we look further?
Symmetric Inequalities
In preparation for upcoming mathematical competitions, I have conducted a relatively deep study into the proof of inequalities, especially n-th degree symmetric inequalities. We find that proofs of symmetric inequalities are quite common in mathematical competitions. For example, given non-negative numbers x, y, z satisfying x+y+z=1, the 1984 IMO asked us to prove: 0 \leq xy+yz+zx-2xyz \leq \frac{7}{27} In last year’s Guangdong Provincial Mathematics Competition preliminary round, there was a similar problem asking us to prove: 9xyz \leq xy+yz+zx \leq \frac{1}{4} (1+9xyz)
In the 41st IMO, given xyz=1, we were asked to prove: \left(x-1+\frac{1}{y}\right)\left(y-1+\frac{1}{z}\right)\left(z-1+\frac{1}{x}\right) \leq 1 And another similar one: \frac{1}{\sqrt{1+8x}}+\frac{1}{\sqrt{1+8y}}+\frac{1}{\sqrt{1+8z}} \geq 1
Simple examples also include the common AM-GM related inequality x^3+y^3+z^3-3xyz \geq 0. There are many more examples, but I will not list them all. Below, I introduce a "physical proof" for such inequalities. Generally speaking, it is not a "beautiful" proof—one might even call it an "ugly proof"—but it is an effective method in many situations.
Unifying Dimensions
Now we treat all unknowns as quantities with the dimension of length, and all constants as dimensionless quantities. In mathematics, we often see expressions like a-1, but in physics, this can never appear because it implies subtracting a dimensionless quantity from a quantity with dimension, which is invalid. We notice that in some of the examples above, such as 4(xy+yz+zx)-9xyz \leq 1, the left side is "the sum of squares of lengths minus the cube of length," while the right side is dimensionless. This is clearly inconsistent. Thus, we must "unify the dimensions." How? By using the given condition x+y+z=1 and letting: \left\{\begin{aligned} x=\frac{a}{a+b+c}\\ y=\frac{b}{a+b+c}\\ z=\frac{c}{a+b+c} \end{aligned}\right.\qquad(a, b, c \text{ are arbitrary positive numbers}) Substituting these into the inequality 4(xy+yz+zx)-9xyz \leq 1 yields: 4(a+b+c)(ab+bc+ca)-9abc \leq (a+b+c)^3 Now we must prove that the above holds for any a, b, c. We can see that every term in the expression now has the dimension of length cubed.
What if the condition is xyz=1? We can let x=\frac{a}{b}, y=\frac{b}{c}, z=\frac{c}{a} or x=\frac{a^2}{bc}, y=\frac{b^2}{ac}, z=\frac{c^2}{ab}, and so on. For proving inequalities, "unifying dimensions" is often an effective step, even if it is not strictly necessary or the simplest path.
Symmetry "Breaking"
This is the core of this article! Symmetric systems are relatively simple to solve because we have an "ultimate weapon"—transforming a symmetric system to make it asymmetric often simplifies the problem. Remember how we first learned to solve quadratic equations? Given xy=p, x+y=q, which is a symmetric system, we let x=a+b, y=a-b. This turns it into the asymmetric a^2-b^2=p, 2a=q. The problem does not become more complex; instead, the solution becomes easier. Similar methods exist for inequalities.
For example, to prove 4(a+b+c)(ab+bc+ca)-9abc \leq (a+b+c)^3, the roles of a, b, c are equivalent. We make them unequal; without loss of generality, let a \geq b \geq c. To further distinguish their roles, we can set: b=c+u, \quad a=b+v=c+u+v where u, v \geq 0.
Substituting these into (a+b+c)^3 - 4(a+b+c)(ab+bc+ca) + 9abc, what do we get? After expansion, we have: \begin{aligned} &(3c+2u+v)^3 \\ &-4(3c+2u+v)[(c+u+v)(c+u)+(c+u)c+(c+u+v)c] \\ &+9(c+u+v)(c+u)c \\ &=v^3+2uv^2+cv^2+cuv+cu^2 \end{aligned}
Clearly, every term is non-negative, so the inequality holds. This is an asymmetric polynomial expansion. The process is quite tedious; as I emphasized, this is not a "beautiful" proof method.
Let’s take another example: proving a^3+b^3+c^3-3abc \geq 0. This involves expanding (c+u+v)^3+(c+u)^3+c^3-3(c+u+v)(c+u)c, which yields: 3cu^2+3cuv+3cv^2+2u^3+3u^2v+3uv^2+v^3 This expression is obviously non-negative, so the inequality holds.
Epilogue
This article is the result of my research into inequalities. Of course, this is not a new concept; I am simply recording it here in the hope that it might be helpful. I have used some terminology from physics because I am a physics enthusiast and believe that physics and mathematics are inseparable. Using physics to understand mathematics can sometimes lead to wonderful results! ^_^ Of course, this article only touches upon the tip of the iceberg of inequality proofs. Furthermore, due to the large amount of calculation involved, it can at best be considered a "machine proof"; for cases with higher degrees, the amount of calculation will be very large and it is not suitable for use.
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