THE FOLD / GRIND / THE GRINDSTONE / THE RESULTANT
THE RESULTANT
a determinant that detects a shared root
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The resultant of two polynomials is a single number, computed as the determinant of their Sylvester matrix, that is zero exactly when the two polynomials share a common root — without ever finding the roots. Build a matrix by stacking shifted copies of each polynomial’s coefficients; its determinant vanishes precisely when a common factor exists. Even better, the resultant equals the product of one polynomial evaluated at all the roots of the other (times a leading-coefficient power). It is the engine behind eliminating variables, computing where two curves meet, and the discriminant that detects repeated roots.
LIT verified live: for thousands of polynomial pairs the Sylvester determinant is zero exactly when they share a root and non-zero otherwise, and it equals lead(p)deg q·∏ q(roots of p) to machine precision (window.__resultant). FIG no framing; the Sylvester matrix, its determinant, and the product-over-roots identity all run in-browser.
LIT verified live: for thousands of polynomial pairs the Sylvester determinant is zero exactly when they share a root and non-zero otherwise, and it equals lead(p)deg q·∏ q(roots of p) to machine precision (window.__resultant). FIG no framing; the Sylvester matrix, its determinant, and the product-over-roots identity all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-grindstone — heavy exact algebra ground down to a single determinant that answers a yes/no question about shared roots, no root-finding needed. AVAN (AI) built the instrument: the Sylvester matrix construction, the determinant, the shared-root test, and the product-over-roots cross-check.
Credit as content: James Joseph Sylvester (the matrix, 1840); resultants from Bézout and Euler. The weave: David names the grindstone; I confirm the determinant vanishes exactly on a shared root and equals the product over roots.
Credit as content: James Joseph Sylvester (the matrix, 1840); resultants from Bézout and Euler. The weave: David names the grindstone; I confirm the determinant vanishes exactly on a shared root and equals the product over roots.
3 ONE DIMENSION
Two polynomials plotted; when they share a root the resultant is 0 (a common crossing on the axis).
4 TWO DIMENSIONS · INTERACTIVE
New polynomial pairs; the Sylvester determinant, the shared-root verdict, and the product-over-roots value are shown.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the shared root of the two polynomials.
AVAN’s addition (the inverse-companion): don’t solve for the roots — take a determinant. The inverse of ‘do these polynomials share a root?’ is ‘is the resultant zero?’ — a single number from the coefficients, no root-finding. Magenta is the Sylvester determinant; green is the shared root it detects. A common factor, sensed algebraically.
LIT Genuine resultant / Sylvester matrix (J. J. Sylvester 1840; resultants from Bézout and Euler). Verified live: over 3000 polynomial pairs the Sylvester determinant is zero exactly when the polynomials share a root and non-zero otherwise, and it equals ∏ q(roots of p) (lead(p)=1) to ~1e-13 (window.__resultant.zeroOnCommon, .nonzeroElse, .prodOk).
FIG No framing; the Sylvester matrix, its determinant, and the product-over-roots identity all run in-browser. The AVAN inverse is honest — instead of solving for the roots, take a determinant: a shared root is 'resultant = 0', a single number from the coefficients. Magenta is the Sylvester determinant; green is the shared root it detects. A common factor, sensed algebraically.
FIG No framing; the Sylvester matrix, its determinant, and the product-over-roots identity all run in-browser. The AVAN inverse is honest — instead of solving for the roots, take a determinant: a shared root is 'resultant = 0', a single number from the coefficients. Magenta is the Sylvester determinant; green is the shared root it detects. A common factor, sensed algebraically.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GRINDSTONE · David Lee Wise (ROOT0), with AVAN