Witt#2: Polynomials that can be written as wn

Witt vectors. Part 1 Michiel Hazewinkel

Sidenotes by Darij Grinberg

Witt#2: Polynomials that can be written as wn

[version 1.0 (15 April 2013), completed, sloppily proofread]

This is an addendum to section 5 of [1]. We recall the definition of the p-adic Witt polynomials:

Definition. Letpbe a prime. For everyn∈N(whereNmeans{0,1,2, ...}), we define a polynomial w_n ∈Z[X₀, X₁, X₂, ..., X_n] by

w_n(X₀, X₁, ..., X_n) = X₀^pn+pX₁^pn−1+p²X₂^pn−2+...+pⁿ⁻¹X_n−1^p +pⁿX_n=

n

X

k=0

p^kX_k^pn−k.

Since Z[X₀, X₁, X₂, ..., X_n] is a subring of the ring Z[X₀, X₁, X₂, ...] (this is the polynomial ring over Z in the countably many indeterminates X₀, X₁, X₂, ...), this polynomial w_n can also be considered as an element of Z[X₀, X₁, X₂, ...]. Regarding w_n this way, we have

w_n(X₀, X₁, X₂, ...) =

n

X

k=0

p^kX_k^pn−k.

We will often write X for the sequence (X₀, X₁, X₂, ...). Thus, w_n(X) =

n

P

k=0

p^kX_k^pn−k.

These polynomials w₀(X), w₁(X), w₂(X), ... are called the p-adic Witt polynomials.¹

A property of these polynomials has not been recorded in the text:

Theorem 1. Letτ ∈Z[X0, X1, X2, ...] be a polynomial. Letn ∈N. Then, the following two assertions A and B are equivalent:

Assertion A: There exist polynomialsτ₀,τ₁,...,τ_ninZ[X₀, X₁, X₂, ...] such that

τ(X) = wn(τ0(X), τ1(X), ..., τn(X)). Assertion B: We have ∂

∂Xi

(τ(X))∈pⁿZ[X₀, X₁, X₂, ...] for every i∈N.

1Caution: These polynomials are referred to as w0, w1, w2, ... in Sections 5-8 of [1]. However, beginning with Section 9 of [1], Hazewinkel uses the notations w1, w2, w3, ... for some different polynomials (the so-called big Witt polynomials, defined by formula (9.25) in [1]), which arenot the same as our polynomials w1, w2, w3, ...(though they are related to them: in fact, the polynomialwk

that we have just defined here is the same as the polynomial which is calledw_pk in [1] from Section 9 on, up to a change of variables; however, the polynomial which is calledwk from in [1] from Section 9 on is totally different and has nothing to do with ourw_k).

Proof of Theorem 1. Proof of the implication A=⇒ B: Assume that Assertion A holds, i. e., that there exist polynomials τ0,τ1, ..., τn in Z[X0, X1, X2, ...] such that

τ(X) = w_n(τ₀(X), τ₁(X), ..., τ_n(X)). Then,

τ(X) =w_n(τ₀(X), τ₁(X), ..., τ_n(X)) =

n

X

k=0

p^k(τ_k(X))^pn−k, so that every i∈N satisfies

∂

∂X_i(τ(X)) = ∂

∂X_i

n

X

k=0

p^k(τ_k(X))^pn−k =

n

X

k=0

p^k ∂

∂X_i (τ_k(X))^pn−k

| {z }

=p^n−k(τk(X))^pn−k−1· ∂

∂X_i^(τk(X))

(by the chain rule, since

∂

∂Y

Y^pn−k

=p^n−kY^pn−k−1)

=

n

X

k=0

p^kp^n−k

| {z }

=pⁿ

(τ_k(X))^pn−k−1· ∂

∂X_i (τ_k(X)) =pⁿ

n

X

k=0

(τ_k(X))^pn−k−1· ∂

∂X_i (τ_k(X))

| {z }

∈Z[X0,X1,X2,...]

∈pⁿZ[X₀, X₁, X₂, ...],

and thus AssertionB holds. This proves the implication A=⇒ B.

Proof of the implication B =⇒ A: Proving the implication B =⇒ A is equivalent to proving the following fact:

Lemma: Let τ ∈ Z[X₀, X₁, X₂, ...] be a polynomial. Let n ∈ N. If ∂

∂Xi

(τ(X)) ∈ pⁿZ[X₀, X₁, X₂, ...] for every i ∈ N, then there exist polynomials τ₀, τ₁, ..., τ_n in Z[X₀, X₁, X₂, ...] such that

τ(X) = wn(τ0(X), τ1(X), ..., τn(X)).

Proof of the Lemma: We will prove the Lemma by induction overn. Forn= 0, the Lemma is trivial (just set τ₀ = τ and use w₀(X) = X₀). Now to the induction step:

Given some n ∈ N such that n ≥ 1, we want to prove the Lemma for this n, and we assume that it is already proven for n−1 instead of n. So let τ ∈Z[X₀, X₁, X₂, ...] be

has a unique representation of this kind). Every i∈Nsatisfies

∂

∂X_i (τ(X)) = ∂

∂X_i





X

(j0,j1,j2,...)∈N^Nfin

t(j0,j1,j2,...)X₀^j0X₁^j1X₂^j2...





= X

(j0,j1,j2,...)∈N^Nfin

t_{(j0,j1,j2,...)}X₀^j0X₁^j1X₂^j2...X_i−1^ji−1 ∂

∂X_iX_i^ji

X_i+1^ji+1...

= X

(j0,j1,j2,...)∈N^Nfin

t_{(j0,j1,j2,...)}X₀^j0X₁^j1X₂^j2...X_i−1^ji−1 j_iX_i^ji−1

X_i+1^ji+1...

= X

(j0,j1,j2,...)∈N^Nfin

j_it_{(j0,j1,j2,...)}X₀^j0X₁^j1X₂^j2...X_i−1^ji−1X_i^ji−1X_i+1^ji+1....

Hence, for every (j₀, j₁, j₂, ...)∈N^Nfin, the coefficient of the polynomial ∂

∂X_i (τ(X)) be- fore the monomialX₀^j0X₁^j1X₂^j2...X_i−1^ji−1X_i^ji−1X_i+1^ji+1...isj_it_{(j0,j1,j2,...)}. Therefore, ∂

∂X_i (τ(X))∈ pⁿZ[X₀, X₁, X₂, ...] rewrites asj_it_{(j0,j1,j2,...)} ∈pⁿZfor every (j₀, j₁, j₂, ...)∈N^Nfin(because a polynomial in Z[X₀, X₁, X₂, ...] lies in pⁿZ[X₀, X₁, X₂, ...] if and only if each of its coefficients lies in pⁿZ). In particular, this yields that

for every (j₀, j₁, j₂, ...)∈N^Nfin satisfying p-t_{(j0,j1,j2,...)}, we have j_ipⁿ ∈Z for every i∈N (1) (because jit(j0,j1,j2,...)∈pⁿZand p-t(j0,j1,j2,...) lead to ji ∈pⁿZ, since p is a prime).

We also notice that

a ≡a^pnmodp for every a∈Z (2)

(since Fermat’s Little Theorem yieldsa^pk ≡ a^pkp

=a^pk+1modpfor everyk ∈N, and thus by induction we get a^p0 ≡a^pnmodp).

Now, define a polynomial ρ∈Z[X₀, X₁, X₂, ...] by

ρ(X) = X

(j0,j1,j2,...)∈N^Nfin; p-t_{(j0,j1,j2,...)}

t_{(j0,j1,j2,...)}X₀^j0pnX₁^j1pnX₂^j2pn...

(this is actually a polynomial because of (1)). Then,

τ(X) = X

(j0,j1,j2,...)∈N^Nfin

t_{(j0,j1,j2,...)}X₀^j0X₁^j1X₂^j2...

= X

(j0,j1,j2,...)∈N^Nfin; p|t_(j

0,j1,j2,...)

t_{(j0,j1,j2,...)}

| {z }

≡0 modp,since p|t_(j

0,j1,j2,...)

X₀^j0X₁^j1X₂^j2...+ X

(j0,j1,j2,...)∈N^Nfin; p-t(j0,j1,j2,...)

t_{(j0,j1,j2,...)}X₀^j0X₁^j1X₂^j2...

≡ X

(j0,j1,j2,...)∈N^Nfin; p|t_(j

0,j1,j2,...)

0X₀^j0X₁^j1X₂^j2...

| {z }

=0

+ X

(j0,j1,j2,...)∈N^Nfin; p-t(j0,j1,j2,...)

t_{(j0,j1,j2,...)}X₀^j0X₁^j1X₂^j2...

= X

(j0,j1,j2,...)∈N^N_fin; p-t_(j

0,j1,j2,...)

t_{(j0,j1,j2,...)}

| {z }

≡t^pn

(j0,j1,j2,...)modp, due to (2)

X₀^j0X₁^j1X₂^j2...

| {z }

=

X₀^j0pnX₁^j1pnX₂^j2pn...pn

(this makes sense becausejipⁿ∈Zfor everyi∈N (by (1), sincep-t_{(j0,j1,j2,...)}))

≡ X

(j0,j1,j2,...)∈N^N_fin; p-t_{(j0,j1,j2,...)}

t^p_(jⁿ

0,j1,j2,...)

X₀^j0pnX₁^j1pnX₂^j2pn...pⁿ

= X

(j0,j1,j2,...)∈N^N_fin; p-t_{(j0,j1,j2,...)}

t_{(j0,j1,j2,...)}X₀^j0pnX₁^j1pnX₂^j2pn...pⁿ

≡





















X

(j0,j1,j2,...)∈N^N_fin; p-t_{(j0,j1,j2,...)}

t_{(j0,j1,j2,...)}X₀^j0pnX₁^j1pnX₂^j2pn...

| {z }

=ρ(X)





















pⁿ





since P

s∈S

a^p_sⁿ ≡

P

s∈S

a_s pⁿ

modp for any family (as)_s∈S of elements of any commutative ring





= (ρ(X))^pnmodp

pⁿ

For every i∈N, we have

p ∂

∂X_i(τe(X)) = ∂

∂X_i





 pτe(X)

| {z }

=τ(X)−(ρ(X))^pn





= ∂

∂X_i

τ(X)−(ρ(X))^pn

= ∂

∂X_i(τ(X))− ∂

∂X_i

(ρ(X))^pn

| {z }

=pⁿ(ρ(X))^pn−1 ∂

∂Xi (ρ(X))

(by the chain rule, since

∂

∂Y (^Ypn)^=pnYpn−1)

= ∂

∂X_i (τ(X))

| {z }

∈pⁿZ[X0,X1,X2,...]

−pⁿ(ρ(X))^pn−1 ∂

∂X_i (ρ(X))

| {z }

∈Z[X0,X1,X2,...]

∈pⁿZ[X₀, X₁, X₂, ...]−pⁿZ[X₀, X₁, X₂, ...]

⊆pⁿZ[X₀, X₁, X₂, ...] (since pⁿZ[X₀, X₁, X₂, ...] is a Z-module), so that

∂

∂X_i (τe(X))∈ 1

ppⁿZ[X₀, X₁, X₂, ...] =pⁿ⁻¹Z[X₀, X₁, X₂, ...].

Therefore, we can apply the Lemma with n−1 instead ofnand witheτ instead ofτ (in fact, the Lemma withn−1 instead ofn is guaranteed to hold by our induction assump- tion), and we obtain that there exist polynomials τe₀, eτ₁, ..., eτn−1 in Z[X₀, X₁, X₂, ...]

such that

eτ(X) = wn−1(eτ₀(X),eτ₁(X), ...,eτn−1(X)). In other words,

eτ(X) = wn−1(eτ₀(X),eτ₁(X), ...,eτn−1(X)) =

n−1

X

k=0

p^k(τe_k(X))^p(n−1)−k.

Now, define polynomials τ₀, τ₁, ...,τ_n in Z[X₀, X₁, X₂, ...] by

τ_k =

ρ, if k = 0;

τek−1, if k >0 for every k ∈ {0,1, ..., n}

.

Then,

w_n(τ₀(X), τ₁(X), ..., τ_n(X))

=

n

X

k=0

p^k(τ_k(X))^pn−k = p⁰

|{z}

=1



 τ₀

|{z}

=ρ

(X)





pⁿ⁻⁰

+

n

X

k=1

p^k

|{z}

=pp^k−1



 τ_k

|{z}

=eτk−1

(X)





p^n−k

= (ρ(X))^pn−0 +

n

X

k=1

pp^k−1 (eτk−1(X))^pn−k

| {z }

=(eτk−1(X))^p(n−1)−(k−1)

= (ρ(X))^pn−0 +

n

X

k=1

pp^k−1(eτ_k−1(X))^p(n−1)−(k−1)

= (ρ(X))^pn−0 +

n−1

X

k=0

pp^k(τe_k(X))^p(n−1)−k (here we substituted k for k−1 in the sum)

= (ρ(X))^pn +p

n−1

X

k=0

p^k(τe_k(X))^p(n−1)−k

| {z }

=eτ(X)

= (ρ(X))^pn+ peτ(X)

| {z }

=τ(X)−(ρ(X))^pn

=τ(X).

This proves our Lemma (i. e., the induction is complete), and thus, the implication B=⇒ A is established.

Altogether, we have proven the implications A=⇒ B and B=⇒ A. Consequently, Assertions A and B are equivalent. Theorem 1 is now proven.

Remark: While it is tempting to believe that our Theorem 1 yields Theorem 5.2 from [1], this doesn’t seem to be the case.²

References

[1] Michiel Hazewinkel, Witt vectors. Part 1, revised version: 20 April 2008.