ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

1.1givenL1L2algebra

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

2.1step 1.1L2algebra

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 $(3, 0)$ .

3.1step 2.1algebra∎

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.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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​