SYMMETRIC FUNCTIONS the building blocks of order-free polynomials — and the coefficients of every equation
The elementary symmetric polynomials are the canonical permutation-invariant functions: e₁ = ∑xi (the sum), e₂ = ∑i<jxixj (sum of pairwise products), up to en = ∏xi (the product). Every symmetric polynomial is a combination of them (the fundamental theorem of symmetric polynomials). And they are exactly Vieta's formulas: the coefficients of a polynomial are the elementary symmetric functions of its roots — so the coefficients cannot tell the roots apart, which is why a quadratic's two roots enter symmetrically. Order-free by construction, and the hinge between roots and coefficients.
THE TECHNIQUE e₁=∑x, e₂=∑xixj, … ; every symmetric poly is a combo of these
The demo computes the elementary symmetric polynomials of {1,2,3} — the Vieta coefficients, invariant under permutation: live demo
HISTORY & CREDIT Newton · Girard · Vieta
“A polynomial's coefficients name its roots.” — they name only the roots' symmetric functions (Vieta), so they cannot tell the roots apart; symmetry is baked into the coefficients. cited
the basis · e₁,e₂,…,en — every symmetric polynomial is a polynomial in them (fundamental theorem). Vieta · a monic polynomial's coefficients are ±ek of its roots. Newton · Newton's identities relate the ek to power sums.
Sum, sum-of-products, product — the order-free basis, and the coefficients of every equation. Symmetry, in the numbers themselves. Vieta / Newton
RECOMMEND FOR I-13 the elementary symmetric polys, on the compiler
On the canonical compiler, for {1,2,3}: e₁=6, e₂=11, e₃=6 — the Vieta coefficients:
$ i13 run sy_symmetricfunction.i13 # elementary symmetric polynomials of {1,2,3}
RUN OK · 30 step(s) · peak stack 3 · call depth 0
e1 = 6 -- sum
e2 = 11 -- sum of pairwise products
e3 = 6 -- product
Recommend: symmetric functions are permutation-invariance made concrete — the order-free basis, and (via Vieta) the coefficients of every polynomial — and i13 computes e₁,e₂,e₃ of {1,2,3}. Not a keeper (they are symmetric by definition — coextensive with permutation-invariance, witnessed). The dart that shows why coefficients cannot tell roots apart: the map from roots to coefficients is symmetric on purpose.