Symmetric Polynomials & Newton’s Sums
Study notes for beginners – no heavy jargon, just plain language.
1. What is a Symmetric Polynomial?
- A polynomial in several variables is symmetric if you can swap any two variables and the polynomial stays the same.
- Everywhere the variables look the same – the polynomial only cares about how many of each value you have, not which variable has which value.
Everyday Analogy
Think of a bag of colored marbles.
If you pick the bag, the order of marbles inside doesn’t matter; only the counts of each color do.
A symmetric polynomial is like a rule that depends only on those counts.
2. Elementary Symmetric Polynomials
For three variables (x, y, z) we have:
| Symbol | Formula | What it “does” |
|---|---|---|
| (e_1) | (x + y + z) | Sum of the variables |
| (e_2) | (xy + yz + zx) | Sum of pairwise products |
| (e_3) | (xyz) | Product of all three |
- These are the basic building blocks.
- Any symmetric polynomial can be rewritten using just (e_1, e_2, e_3).
Tiny Example
If (x=1, y=2, z=3):
- (e_1 = 1+2+3 = 6)
- (e_2 = 1·2 + 2·3 + 3·1 = 2 + 6 + 3 = 11)
- (e_3 = 1·2·3 = 6)
3. Power Sums
Power sums are just sums of powers of the variables:
- (p_1 = x + y + z)
- (p_2 = x^2 + y^2 + z^2)
- (p_3 = x^3 + y^3 + z^3)
- …and so on.
They are useful when you need to know something like “sum of squares” or “sum of cubes.”
4. Newton’s Sums (Newton’s Identities)
Newton’s sums give a bridge between elementary symmetric polynomials ((e_k)) and power sums ((p_k)).
The Simple Rule
For a monic polynomial of degree (n)
[ x^n + a_1 x^{n-1} + a_2 x^{n-2} + \dots + a_{n-1}x + a_n = 0 ]
with roots (r_1, r_2, \dots, r_n),
the Newton sums say:
[ p_k + a_1 p_{k-1} + a_2 p_{k-2} + \dots + a_{k-1} p_1 + k a_k = 0 \quad (k = 1,2,\dots,n) ]
(For (k>n) the same formula holds but with (a_k = 0).)
What the Formula Does
- It lets you compute (p_k) (sum of (k)-th powers) using the known coefficients (a_i).
- The coefficients (a_i) are the elementary symmetric polynomials up to sign.
Everyday Analogy
Imagine you have a recipe book (the polynomial) and a list of ingredients (the (a_i)).
Newton’s sums are like a kitchen calculator that, using the recipe, instantly tells you the total amount of each ingredient you need for a certain number of servings (the power sums).
5. How to Use Newton’s Sums – Step by Step
-
Write the polynomial in standard (monic) form.
Example: (x^3 - 6x^2 + 11x - 6 = 0) -
Identify the coefficients (a_1, a_2, a_3).
Here: (a_1 = -6), (a_2 = 11), (a_3 = -6). -
Set up the first few equations using the rule for (k = 1, 2, 3).
For (k=1):
(p_1 + a_1 = 0 \Rightarrow p_1 = -a_1 = 6)For (k=2):
(p_2 + a_1 p_1 + 2a_2 = 0)
(p_2 + (-6)(6) + 2(11) = 0)
(p_2 - 36 + 22 = 0 \Rightarrow p_2 = 14)For (k=3):
(p_3 + a_1 p_2 + a_2 p_1 + 3a_3 = 0)
(p_3 + (-6)(14) + 11(6) + 3(-6) = 0)
(p_3 - 84 + 66 - 18 = 0 \Rightarrow p_3 = 36) -
Interpret the results.
- (p_1 = x + y + z = 6)
- (p_2 = x^2 + y^2 + z^2 = 14)
- (p_3 = x^3 + y^3 + z^3 = 36)