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Concept

Polynomial

An expression built from a variable by adding and multiplying only, with whole-number powers.

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

012345678910012345678910sentf(0) = 4f(1) = 8f(2) = 7f(3) = 1f(4) = 1f(5) = 7f(6) = 8the message 4, 1, 3 read as 3x² + 1x + 4 over GF(11), evaluated at x = 0…6all 35 ways of keeping 3 of the 7 values were interpolated and every one returned the message

A polynomial through the gaps

Write the message as the coefficients of a polynomial and send its values instead. Any k of them determine the polynomial, so it does not matter which ones are lost — and it does not matter how many, as long as k survive.

computation · error correcting codes radius 0.40 turns0 roots insideradius 10 turns0 roots insideradius 1.42 turns2 roots insideradius 23 turns3 roots insidep(z) = z³ − 2z + 2 — as the circle grows, the number of turns steps up from 0 to 3the count can only change by the image crossing the origin, and a crossing is a root

A loop that cannot miss the middle

Feed a circle into a polynomial and a closed loop comes out. A small circle gives a loop that does not enclose the origin; a large one gives a loop that goes round it as many times as the degree. Something has to happen in between, and that something is a root.

algebra · polynomial roots

Named alongside it

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

Closed curveComplex numbersContinuityDegreeErasureError correcting codeExistence proofFinite fieldFundamental theoremInterpolationMinimum distanceNonconstructive

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