Alphabeta Math

Symmetric Polynomials and the Fundamental Theorem of Symmetric Functions — Examples

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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.

✓ 9 results · all verified · 6 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 3 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

Reducing a symmetric polynomial in two variables to a polynomial in e₁ and e₂

Example

The lexicographic reduction algorithm in three variables

Counterexample

A fixed unbalanced one-variable substitution need not preserve lexicographic leading terms

Example

Newton's identities through p₄ in three variables

Counterexample

Power sums need not generate the symmetric ring in characteristic two

Example

Complete homogeneous symmetric polynomials and their recurrence in two variables

Example

The discriminant of x²+bx+c and its double-root criterion

Example

The discriminant of x³+px+q is -4p³-27q²

Example

Computing the monic resultant of two quadratics from roots and coefficients

Sources