Concept

Resultant

A determinant built from the coefficients of two polynomials that vanishes exactly when they share a root. It turns two equations in two unknowns into one equation in one, which is how elimination works without finding any root.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Two coefficient arrays, one with determinant zero and one without. The Sylvester matrices of two pairs of polynomials drawn as grids of coefficients, one pair sharing a root and one not, with each determinant computed in whole numbers and checked against whether a shared root exists.

A shared root, found without finding it

Two polynomials have a root in common exactly when one determinant built from their coefficients is zero. No root is computed, nothing is approximated, and the same construction turns two equations in two unknowns into one equation in one.

algebra · Polynomial roots

Two curves with five crossings: x⁶ + (44/31)y³ − y and its mirror. Two curves in the positive quadrant, each the zero set of a trinomial, crossing at five marked points.

Five where four were promised

In one unknown, a polynomial with three terms has at most two positive roots, whatever its degree. The natural guess for two equations in two unknowns, each with three terms, was two times two: four. It is wrong. Bertrand Haas found two such equations in 2002 whose curves cross five times in the positive quadrant, all five crowded within a tenth of one corner, and five turned out to be the most there can ever be.

geometry · Euclidean algorithm

Named alongside it

The objects these essays reach for when they reach for this one.

Polynomial Coefficient Counterexample Curve Descartes rule of signs Determinant Discriminant Elimination Fewnomial Linear map Logarithm Root

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