96 1. Higher order Fourier analysis

In analogy with Exercise 1.6.1, we can now define the general notion of

a polynomial map:

Definition 1.6.7. A map φ: H → G between two filtered groups H•,G• is

said to be polynomial if it maps

HKk(H•)

to

HKk(G•)

for each k ≥ 0. The

space of all such maps is denoted Poly(H• → G•).

Since

HKk(H•), HKk(G•)

are groups, we immediately

obtain20

Theorem 1.6.8 (Lazard-Leibman theorem). Poly(H• → G•) forms a group

under pointwise multiplication.

In a similar spirit, we have

Theorem 1.6.9 (Filtered groups and polynomial maps form a category).

If φ: H → G and ψ : G → K are polynomial maps between filtered groups

H•,G•,K•, then ψ ◦ φ: H → K is also a polynomial map.

We can also give some basic examples of polynomial maps. Any constant

map from H to G taking values in G≥0 is polynomial, as is any map φ: H →

G which is a filtered homomorphism in the sense that it is a homomorphism

from H≥i to G≥i for any i ≥ 0.

Now we turn to an alternate definition of a polynomial map. For any h ∈

H and any map φ: H → G Define the multiplicative derivative Δhφ: H → G

by the formula Δhφ(x) :=

φ(hx)φ(x)−1.

Theorem 1.6.10 (Alternate description of polynomials). Let φ: H → G be

a map between two filtered groups H, G. Then φ is polynomial if and only

if, for any i1,...,im ≥ 0, x ∈ H≥0, and hj ∈ H≥ij for j = 1,...,m, one

has Δh1 . . . Δhm φ(x) ∈ G≥i1+···+im .

In particular, from Exercise 1.6.1, we see that a non-classical polynomial

of degree d from one additive group H to another G is the same thing as a

polynomial map from (H, ≤ 1) to (G, ≤ d). More generally, a φ map from

(H, ≤ 1) to a filtered group G• is polynomial if and only if

Δh1 . . . Δhi φ(x) ∈ G≥i

for all i ≥ 0 and x, h1,...,hi ∈ H.

Proof. We first prove the “only if” direction. It is clear (by using 0-

dimensional cubes) that a polynomial map must map H≥0 to G≥0. To

obtain the remaining cases, it suﬃces by induction on m to show that if φ

20From our choice of definitions, this theorem is a triviality, but the theorem is less trivial

when using an alternate but non-trivially equivalent definition of a polynomial, which we will give

shortly. Lazard [La1954] gave a version of this theorem when H was the integers and G was a

nilpotent Lie group; the general problem of multiplying polynomial sequences was considered by

Leibman [Le1998].