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Reducing a symmetric polynomial in two variables to a polynomial in $e_{1}$ and $e_{2}$

Example

For

$f (x, y) = x^{3} + y^{3} + x^{2} y + x y^{2},$

the lexicographic reduction algorithm gives

$f = e_{1}^{3} - 2 e_{1} e_{2},$

where $e_{1} = x + y$ and $e_{2} = x y$ .

Facts & Assumptions

Given: The symmetric polynomial $f$ above over $Z$ .

[L1]

Every symmetric polynomial is a polynomial in the elementary symmetric polynomials, obtained by cancelling its leading monomial lexicographically (Every symmetric polynomial over a commutative ring is a polynomial in the elementary symmetric polynomials).

[L2]

In two variables, $e_{1} = x + y$ and $e_{2} = x y$ (The elementary symmetric polynomials $e_{0}, e_{1}, \dots, e_{n}$ ).

Verification

technique · direct

1.1

The leading monomial of $f$ is $x^{3}$ . The elementary monomial prescribed by [L1] is $e_{1}^{3}$ , and $e_{1}^{3} = x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}$ .

givenL1L2algebra

2.1

Subtracting gives $f - e_{1}^{3} = - 2 x^{2} y - 2 x y^{2} = - 2 x y (x + y) = - 2 e_{1} e_{2}$ , whose leading multidegree is smaller than $(3, 0)$ .

step 1.1L2algebra

3.1

Hence $f = e_{1}^{3} - 2 e_{1} e_{2}$ in $Z [x, y]$ . Both sides have integer coefficients, so expanding the right side recovers the original polynomial over every commutative ring; there the remainder $- 2 e_{1} e_{2}$ of step 2.1 may itself vanish, as it does in characteristic $2$ , in which case the algorithm halts one step earlier at the same answer.

step 2.1algebra∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 13 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

K. Conrad, Symmetric Polynomials, Section 2 (standard reference, not scraped)

Reducing a symmetric polynomial in two variables to a polynomial in e1e1​ and e2e2​