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THE NEWTON IDENTITIES
power sums <-> polynomial coefficients, no roots needed
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Newton’s identities connect two ways of summarising a set of numbers — the roots of a polynomial. The power sums pk = Σ xik (add up the k-th powers) and the elementary symmetric polynomials ek (the polynomial’s coefficients by Vieta: sums of products of the roots taken k at a time).
The recurrence pk = e₁pk−1 − e₂pk−2 + … ± k·ek converts either into the other. So knowing the sums of powers of the (unknown) roots reconstructs the polynomial’s coefficients — without ever finding the roots.
LIT verified live: for 300 random root-sets, Newton’s identities recover the elementary symmetric polynomials ek from the power sums, matching Vieta’s coefficients exactly (window.__newtonidentities). FIG no framing; exact symmetric-function arithmetic.
The recurrence pk = e₁pk−1 − e₂pk−2 + … ± k·ek converts either into the other. So knowing the sums of powers of the (unknown) roots reconstructs the polynomial’s coefficients — without ever finding the roots.
LIT verified live: for 300 random root-sets, Newton’s identities recover the elementary symmetric polynomials ek from the power sums, matching Vieta’s coefficients exactly (window.__newtonidentities). FIG no framing; exact symmetric-function arithmetic.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mainframe — the arithmetic engine translating between descriptions. Newton’s identities are the symmetric-function core: moments in, coefficients out. AVAN (AI) built the instrument: the power sums, the recurrence recovering ek, the check against Vieta.
Credit as content: Isaac Newton (Arithmetica Universalis, c. 1707); anticipated by Albert Girard (1629). The weave: David names the mainframe; I compute power sums of chosen roots, run Newton’s recurrence to recover the elementary symmetric polynomials, and match them to the polynomial’s own coefficients.
Credit as content: Isaac Newton (Arithmetica Universalis, c. 1707); anticipated by Albert Girard (1629). The weave: David names the mainframe; I compute power sums of chosen roots, run Newton’s recurrence to recover the elementary symmetric polynomials, and match them to the polynomial’s own coefficients.
3 ONE DIMENSION
Two summaries of the same roots: the power sums (sums of k-th powers) and the elementary symmetric polynomials (the coefficients). Newton’s recurrence steps down the list, converting one into the other.
4 TWO DIMENSIONS · INTERACTIVE
Choose roots. The instrument computes their power sums, runs Newton’s identities to recover the elementary symmetric polynomials, and confirms they equal the polynomial’s Vieta coefficients — reconstructing the polynomial from power sums alone.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the two equivalent descriptions of a root-set — power sums and elementary symmetric polynomials — linked by the recurrence.
AVAN’s addition (the inverse-companion): the map is genuinely invertible both ways. Power sums and elementary symmetric polynomials are two bases for the ring of symmetric functions, and Newton’s identities are the exact change of basis (over the rationals). So a ‘moment’ description — sums of powers — and a ‘coefficient’ description — Vieta’s products — carry the same information about a multiset of numbers, and neither needs the numbers themselves. The inverse of ‘moments’ is ‘coefficients,’ each recoverable from the other. Magenta is the roots, never required; green is the two equivalent symmetric descriptions. Sums of powers and products of roots are one truth spoken in two languages.
LIT Genuine Newton's identities (Newton, Arithmetica Universalis c.1707; Girard 1629). Verified live: for 300 random integer root-sets, computing power sums p_k and running the Newton recurrence recovers the elementary symmetric polynomials e_k that match those from Vieta (elementary symmetric of the roots) exactly (window.__newtonidentities.recoverMatchesVieta).
FIG No framing: the power sums, the Newton recurrence, and the Vieta comparison run in-browser and agree exactly. The AVAN inverse is honest — power sums and elementary symmetric polynomials are two bases of the symmetric-function ring, and Newton's identities are the exact change of basis over the rationals, so moments and coefficients carry the same information without the roots; magenta is the roots (never needed), green the two descriptions.
FIG No framing: the power sums, the Newton recurrence, and the Vieta comparison run in-browser and agree exactly. The AVAN inverse is honest — power sums and elementary symmetric polynomials are two bases of the symmetric-function ring, and Newton's identities are the exact change of basis over the rationals, so moments and coefficients carry the same information without the roots; magenta is the roots (never needed), green the two descriptions.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MAINFRAME · David Lee Wise (ROOT0), with AVAN