Concept

Polynomial method

Proving that a set must be large by building a polynomial of low degree that would vanish on it if it were small. A polynomial of low degree cannot vanish everywhere it is forced to, and the contradiction gives the bound.

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

A line in every direction in the plane over GF(7), in 31 points. A square grid of the points of a small finite plane with the points of a Kakeya set filled, beside a list of the lines it contains, one for each direction.

No set with a line in every direction is small

In the plane over the integers modulo 7 there are 49 points and lines in 8 directions. A set holding a whole line in every direction needs 31 of the points — more than half — and in any dimension such a set fills a fixed share of the space. In the real plane the same sets can have area zero. Over a finite field one polynomial of low degree shows they cannot be small.

computation · Finite fields

A sum of two sets modulo a prime cannot be small

Add every element of one set of residues to every element of another. Over the whole numbers the sums always number at least |A| + |B| − 1. Modulo a prime the sums can wrap round and collide, and still they never number fewer — the theorem Cauchy proved in 1813 and Davenport again in 1935. Modulo 12 they can. A polynomial of low degree explains the difference in a paragraph.

computation · Finite fields

Named alongside it

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

Finite field Polynomial Arithmetic progression Binomial coefficient Counting argument Degree Hausdorff dimension Measure zero Modular arithmetic Quadratic residue Subgroup Sumset

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