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THE DESCARTES
bound the positive roots by counting sign changes
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Descartes’ rule of signs reads a bound on a polynomial’s positive real roots straight off its coefficients: the number of positive roots is at most the number of sign changes in the coefficient sequence, and differs from it by an even number. (Substituting x→−x gives the same bound for negative roots.) You learn something about the roots before computing any of them.
LIT verified live: over 300 polynomials built from known real roots, the true count of positive roots is always ≤ the sign-change count and has the same parity (window.__descartes). FIG no framing; exact bound.
LIT verified live: over 300 polynomials built from known real roots, the true count of positive roots is always ≤ the sign-change count and has the same parity (window.__descartes). FIG no framing; exact bound.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mainframe — the heavy algebra a mainframe runs, but here a bound on the roots is read for free from the signs, no solving required. Descartes’ rule is that free look. AVAN (AI) built the instrument: the root-to-coefficient expansion, the sign-change counter, the bound-and-parity check.
Credit as content: René Descartes (1637). The weave: David names the mainframe; I build polynomials from chosen roots, count sign changes, and confirm the positive-root count never exceeds it and shares its parity.
Credit as content: René Descartes (1637). The weave: David names the mainframe; I build polynomials from chosen roots, count sign changes, and confirm the positive-root count never exceeds it and shares its parity.
3 ONE DIMENSION
Walk the coefficients from highest to lowest degree, skipping zeros; each time the sign flips, count one. That count bounds the positive real roots (and the shortfall is even).
4 TWO DIMENSIONS · INTERACTIVE
A polynomial (built from known roots); its sign changes and true positive-root count are shown — the count ≤ sign changes, same parity.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the sign-change count that bounds the positive roots.
AVAN’s addition (the inverse-companion): the number of positive real roots is bounded by the number of sign changes in the coefficient sequence — and differs from it by an even number — so you read a bound straight off the coefficients, before finding any root. The inverse of ‘solve for the roots then count’ is ‘count sign changes in the coefficients — the positive roots can’t exceed that, same parity.’ Magenta is the roots you don’t compute; green is the sign-change count that bounds them. The coefficients already whisper how many positive roots there are.
LIT Genuine Descartes' rule of signs (Descartes 1637). Verified live: for 300 polynomials built from known real roots, the count of positive roots is always <= the number of coefficient sign changes and has the same parity (the shortfall is even) (window.__descartes.boundAndParity).
FIG No framing: the root-to-coefficient expansion, the sign-change counter, and the bound-and-parity check run in-browser and hold exactly. The AVAN inverse is honest — the positive-root count is bounded by (and shares parity with) the coefficient sign-change count, read off before solving; magenta is the roots not computed, green the sign-change bound. The coefficients whisper how many positive roots there are.
FIG No framing: the root-to-coefficient expansion, the sign-change counter, and the bound-and-parity check run in-browser and hold exactly. The AVAN inverse is honest — the positive-root count is bounded by (and shares parity with) the coefficient sign-change count, read off before solving; magenta is the roots not computed, green the sign-change bound. The coefficients whisper how many positive roots there are.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MAINFRAME · David Lee Wise (ROOT0), with AVAN