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Concept

Coefficient

One of the numbers multiplying the powers of the variable in a polynomial, listed in order of that power. A great deal about the roots — their sum, their product, whether two polynomials share one — is a computation on the coefficients with no root ever found.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

3 roots, and the two numbers the coefficients already knew. The roots of a degree-3 polynomial, found numerically, with the point they average to. That average, and their product, are readable straight off the coefficients without finding the roots at all.

What the coefficients already know

Finding the roots of a polynomial is hard and often impossible in closed form. Reading off their sum, their product and how many of them are real is none of those things — those numbers are sitting in the coefficients, and no root-finding is required to get at them.

algebra · Polynomial roots 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 The partition product's coefficients to q¹². A row of series coefficients computed by expanding a product, beside the same numbers obtained another way.

Every partition, hidden in a product

Multiply out one factor for each part size and the coefficient of q to the n is the number of partitions of n. Nothing is being approximated: the product is a bookkeeping device that does the counting by multiplying.

number · Partitions

Named alongside it

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

DiscriminantPolynomialSymmetric functionAlgebraic identityBijectionComplex numbersCounting two waysCurveDegreeDeterminantEliminationFactorisation

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