Annihilating polynomials for quadratic forms and Stirling numbers of the second kind

Stefan De Wannemacker

August 4, 2005

Abstract

We present a set of generators of the full annihilator ideal for the Witt ring of an arbitrary field of characteristic unequal to two satisfying a nonvanishing condition on the powers of the fundamental ideal in the torsion part of the Witt ring. This settles a conjecture of Ongenae and Van Geel. This result could only be proved by first obtaining a new lower bound on the 2-adic valuation of Stirling numbers of the second kind.

1 Introduction

In 1937, Witt already observed that the Witt ring was integral in the sense that each element was annihilated by a monic integer polynomial. Fifty years later, in 1987, Lewis was the first to give explicit examples of such polynomials [4]. He showed that the monic polynomial, p_{n}(X), defined as

p_{n}(X) = (X − n)(X − (n − 2)) … (X + (n − 2))(X + n)

annihilates every non-singular quadratic form of dimension n over every field F of characteristic unequal to two. Since then, many other polynomials in ℤ[X] were found annihilating all or a family of classes of nonsingular quadratic forms in the Witt ring and we refer the reader to [3] for a nice survey of the main results on this topic.

Let F be a field of characteristic not 2. The object we want to consider here is the torsion annihilator ideal

A_{t}(F) = {f(X) ∈ ℤ[X] | f(ϕ) = 0, ∀ ϕ ∈ I_{t}(F)}

where I_{t}(F) = W_{t}(F) ∩ I(F), W_{t}(F) the torsion part of the Witt ring and I(F) the ideal of all even-dimensional forms in the Witt ring. Since A_{t}(F) is an ideal in the noetherian ring ℤ[X], it is finitely generated. The main problem is to find a set of generators for this ideal. We will prove the following result.

For fields F satisfying the conditions that 2^{r}W_{t}(F) = 0 and 2^{r−1}(I_{t}(F))^{2k−1} ≠ 0 with k uniquely determined by r, the torsion annihilator ideal A_{t}(F) is the ideal generated by the monomials

{2^{r−ν_{2}((2i)!)}X^{2i}}_{0≤i≤k−1} ∪ {X^{2k}},

where ν_{2} denotes the 2-adic valuation function.

In the case of a nonreal field F this theorem was conjectured (see Corollary 3.4 ) by Ongenae and Van Geel [5]. They gave a proof for fields with level s(F) ≤ 16 and using the same technique, you can check that the theorem holds for all nonreal fields F with level s(F) ≤ 64, but a general method was lacking. The general method, used to prove the theorem, consists in evaluating a polynomial f(X) ∈ ℤ[X] in the even-dimensional forms ⊥_{i=1}^{n}⟨⟨a_{i}⟩⟩ with 1 ≤ n ≤ deg(f). This evaluation can be rewritten as a linear combination of sums of m-fold Pfister forms and the coefficients that appear turn out to be related to the Stirling numbers of the second kind. The result about the (torsion) annihilator ideal could only be proved by first obtaining a new lower bound for the 2-adic valuation of all Stirling numbers S(n, k) of the second kind, namely

ν_{2}(S(n, k)) ≥ d(k) − d(n), for 0 ≤ k ≤ n

where d(k) the sum of the binary digits in the binary representation of k.

2 Stirling numbers of the second kind

2.1 Preliminaries

Let n ∈ ℕ . The Stirling numbers S(n, k) (k ∈ ℕ) of the second kind are given by

x^{n} = ∑_{k=0}^{∞} S(n, k)(x)_{k},

where (x)_{k} = x(x−1)(x−2) … (x−k+1) for k ∈ ℕ\{0} and (x)_{0} = 1. Actually S(n, k) is the number of ways in which it is possible to partition a set with n elements in k classes. The Stirling numbers of the second kind can be defined in several ways.

Proposition 2.1.

S(n, k) = \frac{1}{k!} ∑_{i=0}^{k}(−1)^{i}\binom{k}{k − i}(k − i)^{n},
S(n, k) = S(n − 1, k − 1) + kS(n − 1, k)
with S(n, 0) = S(0, k) = 0 and S(0, 0) = 1
S(n, k) = \frac{1}{k!} ∑_{n_{1},n_{2},...,n_{k}} \binom{n}{n_{1}, n_{2}, ...n_{k}},
where n_{1}, n_{2}, ..., n_{k} are non-zero and their sum equals n.

Proof. See [1] and [6].

2.2 2-adic valuation of Stirling numbers of the second kind

The 2-adic valuation of Stirling numbers of the second kind and other combinatorial numbers has been widely studied, but many problems in this area are still unsolved. We will give a new lower bound for the 2-adic valuation of all Stirling numbers of the second kind.

Denote by d(n) the sum of the digits in the binary representation of n and define the 2-adic valuation function ν_{2}(n) for all non-zero integers n by ν_{2}(n) = p, where 2^{p}|n and 2^{p+1} ∤ n. Recall the following properties.

ν_{2}(n!) = n − d(n)  (Legendre)
ν_{2} (\binom{n}{k}) = d(k) + d(n − k) − d(n)  (Kummer)

for all k, n ∈ ℕ with 0 ≤ k ≤ n. A new lower bound on the 2-adic valuation of Stirling numbers of the second kind can be obtained as follows.

Theorem 2.2. Let n, k ∈ ℕ and 0 ≤ k ≤ n. Then

ν_{2} (S(n, k)) ≥ d(k) − d(n).

Proof.

By induction on n.

For n = 0, ν_{2}(S(0, 0)) = ν_{2}(1) ≥ d(0) − d(0).

Assume now that the above inequality is true for all i < n. We will prove the theorem for n. Note that for k = 0 the result is obviously true. Let 1 ≤ k ≤ n. The Stirling numbers of the second kind satisfy the well-known ’vertical’ recurrence relation

S(n, k) = ∑_{i=k−1}^{n−1} \binom{n − 1}{i}S(i, k − 1).

Combining this with the ’triangular’ recurrence relation

S(n, k) = S(n − 1, k − 1) + kS(n − 1, k)

we obtain

kS(n, k) = ∑_{i=k−1}^{n−1} \binom{n}{i}S(i, k − 1).

Thus

ν_{2}(kS(n, k)) = ν_{2} ( ∑_{i=k−1}^{n−1} \binom{n}{i}S(i, k − 1))

≥ min_{k−1≤i≤n−1}\{ν_{2} (\binom{n}{i}) + d(k − 1) − d(i)\}

(by the induction hypothesis)

= min_{k−1≤i≤n−1}{d(n − i) + d(k − 1) − d(n)}

(by the Kummer identity)

= d(k − 1) − d(n) + 1.

So,

ν_{2}(S(n, k)) ≥ d(k − 1) − ν_{2}(k) + 1 − d(n)
= d(k) − d(n).

2.3 Relationship between Stirling numbers of the second kind and quadratic forms

We can evaluate

f(X) = c_{d}X^{d} + … + c_{1}X + c_{0} ∈ ℤ[X]

in classes of quadratic forms ϕ ∈ W (F) by defining

f(ϕ) = c_{d}ϕ^{d} ⊥ … ⊥ c_{1}ϕ ⊥ c_{0}⟨1⟩ ∈ W (F)

where c_{i}ϕ = sign(c_{i}) \underbrace{(ϕ ⊥ … ⊥ ϕ)}_{|c_{i}| times} and ϕ^{i} = \underbrace{ϕ ⊗ … ⊗ ϕ}_{i times}.

For arbitrary k > 0 and a_{1}, a_{2}, …, a_{k} ∈ F^{∗} = F \{0} denote with ⟨⟨a_{1}, a_{2}, …, a_{k}⟩⟩ the 2^{k}-dimensional k-fold Pfister form

⟨⟨a_{1}, a_{2}, …, a_{k}⟩⟩ := ⟨1, a_{1}⟩ ⊗ ⟨1, a_{2}⟩ ⊗ ⟨1, a_{k}⟩.

Stirling numbers of the second kind turn up in a natural way by making calculations in the Witt ring.

Proposition 2.3. Let f(X) = c_{d}X^{d} + … + c_{1}X ∈ ℤ[X].

Then

f(⊥_{i=1}^{n}⟨⟨a_{i}⟩⟩) = (∑_{q=1}^{d} 2^{q−1}1! S(q, 1)c_{q}) (⊥_{i=1}^{n}⟨⟨a_{i}⟩⟩)

⊥ (∑_{q=2}^{d} 2^{q−2}2! S(q, 2)c_{q}) (⊥_{i<j}^{n}⟨⟨a_{i}, a_{j}⟩⟩)

⊥ …

⊥ (∑_{q=n}^{d} 2^{q−n}n! S(q, n)c_{q}) ⟨⟨a_{1}, …, a_{n}⟩⟩.

Proof.

The evaluation of the polynomial f(X) = c_{d}X^{d} + … + c_{1}X in sums of 1-fold Pfister forms ⊥_{i=1}^{n}⟨⟨a_{i}⟩⟩ can be written in the following way.

f(⊥_{i=1}^{n}⟨⟨a_{i}⟩⟩) = A_{1}(c_{1}, …, c_{d})(⊥_{i=1}^{n}⟨⟨a_{i}⟩⟩)
⊥ A_{2}(c_{1}, …, c_{d})(⊥_{i<j}^{n}⟨⟨a_{i}, a_{j}⟩⟩)
⊥ …
⊥ A_{n}(c_{1}, …, c_{d})⟨⟨a_{1}, …, a_{n}⟩⟩,

where the A_{p}(c_{1}, …, c_{d}) are expressions in c_{1}, …, c_{d} with natural numbers as coefficients, i.e.

A_{p}(c_{1}, …, c_{d}) = ∑_{q=1}^{d} γ_{p,q}c_{q}, with γ_{p,q} ∈ ℕ.

The natural number γ_{p,q} is the coefficient of ⟨⟨a_{1}, …, a_{p}⟩⟩ in (⊥_{i=1}^{p}⟨⟨a_{i}⟩⟩)^{q}.

For p ≤ q we can write

γ_{p,q} = ∑_{q_{1},q_{2},...,q_{p}≥1}^{q_{1}+q_{2}+...+q_{p}=q} \binom{q}{q_{1}, q_{2}, ..., q_{p}}2^{q_{1}−1}2^{q_{2}−1}... 2^{q_{p}−1}

= 2^{q−p} ∑_{q_{1},q_{2},...,q_{p}≥1}^{q_{1}+q_{2}+...+q_{p}=q} \binom{q}{q_{1}, q_{2}, ..., q_{p}}

= 2^{q−p}p! S(q, p).

If p > q then clearly no p-fold Pfister form can occur in (⊥_{i=1}^{p}⟨⟨a_{i}⟩⟩)^{q}. So,

γ_{p,q} = \begin{cases}2^{q−p}p! S(q, p) & if p ≤ q, \\ 0 & otherwise. \end{cases}

Applying this to the coefficients A_{p}(c_{1}, …, c_{d}) we obtain

f(⊥_{i=1}^{n}⟨⟨a_{i}⟩⟩) = (∑_{q=1}^{d} 2^{q−1}1! S(q, 1)c_{q}) (⊥_{i=1}^{n}⟨⟨a_{i}⟩⟩)

⊥ (∑_{q=2}^{d} 2^{q−2}2! S(q, 2)c_{q}) (⊥_{i<j}^{n}⟨⟨a_{i}, a_{j}⟩⟩)

⊥ …

⊥ (∑_{q=n}^{d} 2^{q−n}n! S(q, n)c_{q}) ⟨⟨a_{1}, …, a_{n}⟩⟩.

Corollary 2.4. Let f(X) = c_{d}X^{d}+…+c_{1}X+c_{0} ∈ ℤ[X] and ϕ ≃ ⟨a_{1}, a_{2}, … a_{n}⟩ a quadratic form of dimension n. If f(n) := f(n⟨1⟩) = 0 then

f(ϕ) = (∑_{q=1}^{d} ∑_{t=q}^{d} 2^{q−1}1! S(q, 1)\binom{t}{q}(−n)^{t−q}c_{t}) (⊥_{i=1}^{n}⟨⟨a_{i}⟩⟩)

⊥ (∑_{q=2}^{d} ∑_{t=q}^{d} 2^{q−2}2! S(q, 2)\binom{t}{q}(−n)^{t−q}c_{t}) (⊥_{i<j}^{n}⟨⟨a_{i}, a_{j}⟩⟩)

⊥ …

⊥ (∑_{q=n}^{d} ∑_{t=q}^{d} 2^{q−n}n! S(q, n)\binom{t}{q}(−n)^{t−q}c_{t}) ⟨⟨a_{1}, …, a_{n}⟩⟩

Proof. Note that for all

ϕ ≃ ⟨a_{1}, a_{2}, … a_{n}⟩

we have

ϕ ≃ ⊥_{i=1}^{n}⟨⟨a_{i}⟩⟩ − n⟨1⟩.

We can also rewrite f(X) as

f(X) = g(X − n) = ∑_{q=1}^{d} ∑_{t=q}^{d} \binom{t}{q}(−n)^{t−q}c_{t}(X − n)^{q}.

The condition f(n) = 0 implies that the constant term in g(Y) with Y = X − n vanishes. The result follows from the previous proposition applied to g(Y).

Lemma 2.5. Let f(X) = c_{d}X^{d} + … + c_{1}X ∈ ℤ[X]. If f(⊥_{i=1}^{k}⟨⟨a_{σ(i)}⟩⟩) = 0, for all 1 ≤ k ≤ n, σ ∈ S_{d} then

(∑_{q=n}^{d} 2^{q−n}n! S(q, n)c_{q}) ⟨⟨a_{τ(1)}, … a_{τ(n)}⟩⟩ = 0, for all τ ∈ S_{d}.

Proof. By induction on n. For n = 1,

0 = f(⟨⟨a_{i}⟩⟩)

= c_{d}⟨⟨a_{i}⟩⟩^{d} + c_{d−1}⟨⟨a_{i}⟩⟩^{d−1} + … + c_{1}⟨⟨a_{i}⟩⟩

= (2^{d−1}c_{d} + 2^{d−2}c_{d−1} + … + c_{1})⟨⟨a_{i}⟩⟩

= (∑_{q=1}^{d} 2^{q−1}1! S(q, 1)c_{q}) ⟨⟨a_{i}⟩⟩

since S(q, 1) = 1 for all q ≥ 1. Assume now that the lemma is true for all i < n. We will prove the lemma for n. Let σ ∈ S_{d}.

0 = f(⊥_{i=1}^{n}⟨⟨a_{σ(i)}⟩⟩)

= (∑_{q=1}^{d} 2^{q−1}1! S(q, 1)c_{q}) (⊥_{i=1}^{n}⟨⟨a_{σ(i)}⟩⟩)

⊥ (∑_{q=2}^{d} 2^{q−2}2! S(q, 2)c_{q}) (⊥_{i<j}^{n}⟨⟨a_{σ(i)}, a_{σ(j)}⟩⟩)

⊥ …

⊥ (∑_{q=n}^{d} 2^{q−n}n! S(q, n)c_{q}) ⟨⟨a_{σ(1)}, …, a_{σ(n)}⟩⟩

by Proposition 2.3. Note that for every subset U ⊂ {1, …, d} of k ≤ d elements, there exists a permutation τ ∈ S_{d} such that U = {τ(1), …, τ(k)}. So,

0 = (∑_{q=1}^{d} 2^{q−1}1! S(q, 1)c_{q}) (⊥_{τ∈X_{1}⊂S_{d}}⟨⟨a_{τ(1)}⟩⟩)

⊥ (∑_{q=2}^{d} 2^{q−2}2! S(q, 2)c_{q}) (⊥_{τ∈X_{2}⊂S_{d}}⟨⟨a_{τ(1)}, a_{τ(2)}⟩⟩)

⊥ …

⊥ (∑_{q=n}^{d} 2^{q−n}n! S(q, n)c_{q}) ⟨⟨a_{σ(1)}, …, a_{σ(n)}⟩⟩

= (∑_{q=n}^{d} 2^{q−n}n! S(q, n)c_{q}) ⟨⟨a_{σ(1)}, …, a_{σ(n)}⟩⟩

by the induction hypothesis.

3 Polynomials annihilating the Witt Ring

3.1 Preliminaries

The fundamental ideal I(F) of W (F) is the ideal consisting of all even dimensional forms of W (F). Put I_{t}(F) := I(F) ∩ W_{t}(F). A field F is called nonreal if −1 is a sum of squares in F. The natural number

s(F) := min{n ∈ ℕ a_{1}^{2} + a_{2}^{2} + … + a_{n}^{2} = −1, a_{i} ∈ F }

is called the level of F. A field F is called (formally) real if −1 is not a sum of squares in F. In this case we define the level s(F) = ∞. It is a well-known fact that s(F) is a power of two, if F is nonreal ([7] or [2]). The height of F is the smallest two-power h(F) = 2^{d} such that 2^{d}W_{t}(F) = 0. If no such power exists then h(F) := ∞. If F is nonreal then h(F) = 2s(F).

We define the torsion annihilator ideal A_{t}(F) in ℤ[X] by

A_{t}(F) = {f(X) ∈ ℤ[X] f(φ) = 0 for all φ ∈ I_{t}(F)}.

For F nonreal, define the full annihilator ideal A(F) in ℤ[X] by

A(F) = {f(X) ∈ ℤ[X] f(φ) = 0 for all φ ∈ W (F)},

the even annihilator ideal by

A_{e}(F) = {f(X) ∈ ℤ[X] f(φ) = 0 for all even dimensional φ ∈ W (F)}

and the odd annihilator ideal by

A_{o}(F) = {f(X) ∈ ℤ[X] f(φ) = 0 for all odd dimensional φ ∈ W (F)}.

In what follows, let k = k(r) be the natural number uniquely determined by ν_{2}((2k − 2)!) < r ≤ ν_{2}((2k)!) (see [5]).

We define the ideals

J_{e,r} = ({2^{r−ν_{2}((2i)!)}X^{2i}}_{0≤i≤k−1}) + (X^{2k})
= (2^{r}, 2^{r−1}X^{2}, 2^{r−3}X^{4}, ..., 2^{r−ν_{2}((2i)!)}X^{2i}, ..., 2^{r−ν_{2}((2k−2)!)}X^{2k−2}, X^{2k}),
J_{o,r} = ({2^{r−ν_{2}((2i)!)}(X − 1)^{2i}}_{0≤i≤k−1}) + ((X − 1)^{2k})
= (2^{r}, 2^{r−1}(X − 1)^{2}, 2^{r−3}(X − 1)^{4}, ..., 2^{r−ν_{2}((2i)!)}(X − 1)^{2i}, ..., 2^{r−ν_{2}((2k−2)!)}(X − 1)^{2k−2}, (X − 1)^{2k})

and

J_{r} = ({2^{r−ν_{2}((2i)!)}X^{2i}(X − 1)^{2i}}_{0≤i≤k−1}) + (X^{2k}(X − 1)^{2k})
= (2^{r}, 2^{r−1}X^{2}(X − 1)^{2}, 2^{r−3}X^{4}(X − 1)^{4}, ...,

2^{r−ν_{2}((2i)!)}X^{2i}(X − 1)^{2i}, ..., 2^{r−ν_{2}((2k−2)!)}X^{2k−2}(X − 1)^{2k−2}, X^{2k}(X − 1)^{2k}).

Example 3.1.

J_{e,1} = (2, X^{2}),
J_{e,2} = (4, 2X^{2}, X^{4}),
J_{e,3} = (8, 4X^{2}, X^{4}),
J_{e,4} = (16, 8X^{2}, 2X^{4}, X^{6}),
J_{e,5} = (32, 16X^{2}, 4X^{4}, 2X^{6}, X^{8}).
J_{e,6} = (64, 32X^{2}, 8X^{4}, 4X^{6}, X^{8}).

3.2 Generators for the full annihilator ideal

Since ℤ[X] is noetherian, the ideals A_{e}(F), A_{o}(F), A(F) and A_{t}(F) are finitely generated. Under certain conditions we give a set of generators. The following lemma is proved in [5]. We will give an alternative proof, using Stirling numbers of the second kind.

Lemma 3.2. Let F be a field for which 2^{r}W_{t}(F) = 0 then

J_{e,r} ⊆ A_{t}(F).

Proof. Let f(X) = 2^{r−ν_{2}((2i)!)}X^{2i} be one of the generators of J_{e,r} and ϕ ≃ ⟨a_{1}, a_{2}, …, a_{n}⟩ an arbitrary even dimensional element of W_{t}(F). Since n is even, we have f(n⟨1⟩) = 2^{r−ν_{2}((2i)!)+2i} (\frac{n}{2})^{2i} ⟨1⟩ = 2^{r+d(2i)} (\frac{n}{2})^{2i} ⟨1⟩ = 0. By Corollary 2.4

f(ϕ) = (∑_{q=1}^{d} 2^{q−1}1! S(q, 1)\binom{2i}{q}(−n)^{2i−q}2^{r−ν_{2}((2i)!)}) (⊥_{i=1}^{n}⟨⟨a_{i}⟩⟩)

⊥ (∑_{q=2}^{d} 2^{q−2}2! S(q, 2)\binom{2i}{q}(−n)^{2i−q}2^{r−ν_{2}((2i)!)}) (⊥_{i<j}^{n}⟨⟨a_{i}, a_{j}⟩⟩)

⊥ …

⊥ (∑_{q=n}^{d} 2^{q−n}n! S(q, n)\binom{2i}{q}(−n)^{2i−q}2^{r−ν_{2}((2i)!)}) ⟨⟨a_{1}, …, a_{n}⟩⟩.

For all j ≤ q we have

ν_{2} (2^{q−j}j! S(q, j)\binom{2i}{q}(−n)^{2i−q}2^{r−ν_{2}((2i)!)})

= q − j + ν_{2}(j!) + ν_{2}(S(q, j)) + ν_{2} (\binom{2i}{q})

+ (2i − q)ν_{2}(n) + r − ν_{2}((2i)!)

= q − j + j − d(j) + ν_{2}(S(q, j)) + d(q) + d(2i − q) − d(2i)

+ (2i − q)ν_{2}(n) + r − 2i + d(2i)

(by Kummer and Legendre)

≥ q − d(j) + d(j) − d(q) + d(q) + d(2i − q)

+ (2i − q)ν_{2}(n) + r − 2i

(by Theorem 2.2)

≥ q + d(2i − q) + (2i − q) + r − 2i (since n is even)

≥ r

Since 2^{r}W_{t}(F) = 0 it follows that

f(ϕ) = 0

or equivalently that

f(X) ∈ A_{t}(F).

This brings us to the main result of this paper:

Theorem 3.3. Let F be a field such that 2^{r}W_{t}(F) = 0 and 2^{r−1}(I_{t}(F))^{2k−1} ≠ 0 with k uniquely determined by ν_{2}((2k − 2)!) < r ≤ ν_{2}((2k)!) . Then

J_{e,r} = A_{t}(F).

Proof.

Let F be a field such that 2^{r}W_{t}(F) = 0 and 2^{r−1}(I_{t}(F))^{2k−1} ≠ 0. Let k be the unique natural number such that ν_{2}((2k − 2)!) < r ≤ ν_{2}((2k)!).

Since X^{2k} ∈ J_{e,r} annihilates every even quadratic form, we will have to prove that every polynomial of degree 2k − 1,

f(X) = c_{2k−1}X^{2k−1} + ... + c_{1}X + c_{0}, with c_{i} ∈ ℤ,

annihilating every even quadratic torsion form, lies in J_{e,r}.

I_{t}(F) is generated by the elements ⟨⟨a⟩⟩ ∈ I_{t}(F) . The condition on the power of the fundamental ideal implies the existence of elements a_{1}, …, a_{2k−1} ∈ F^{∗} such that the form 2^{r−1}⟨⟨a_{1}, …, a_{2k−1}⟩⟩ is not zero. Fix such elements.

We will evaluate the polynomial f(X) in the even quadratic forms of the form

⊥_{i=1}^{n}⟨⟨a_{σ(i)}⟩⟩

where 1 ≤ n ≤ 2k − 1, σ ∈ S_{2k−1}.

Since f(0) = c_{0}⟨1⟩ has to vanish and 2^{r}W_{t}(F) = 0, it follows that c_{0} = b_{0}2^{r} for some b_{0} ∈ ℤ. By Lemma 2.5 we get the following set of equations in the Witt ring:

( ∑_{q=n}^{2k−1} 2^{q−n}n! S(q, n)c_{q}) ⟨⟨a_{1}, … a_{n}⟩⟩ = 0, 1 ≤ n ≤ 2k − 1.  (1)

For n = 2k − 1, this becomes

0 =(2k − 1)! S(2k − 1, 2k − 1)c_{2k−1}⟨⟨a_{1}, …, a_{2k−1}⟩⟩
=(2k − 1)! c_{2k−1}⟨⟨a_{1}, …, a_{2k−1}⟩⟩.

Since

2^{r−1}⟨⟨a_{1}, …, a_{2k−1}⟩⟩ ≠ 0,

and 2^{r}W_{t}(F) = 0 it follows that

(2k − 1)! c_{2k−1} = b_{2k−1}2^{r} for some b_{2k−1} ∈ ℤ,

and since ν_{2}((2k − 1)!) = ν_{2}((2k − 2)!) < r that

c_{2k−1} = 2^{r−ν_{2}((2k−1)!)}b_{2k−1} for some b_{2k−1} ∈ ℤ.

For n = 2k − 2 , (1) becomes

((2k − 2)! c_{2k−2} + 2(2k − 2)! S(2k − 1, 2k − 2)c_{2k−1})⟨⟨a_{1}, …, a_{2k−1}⟩⟩ = 0. (2)

The second term 2(2k − 2)!S(2k − 1, 2k − 2)c_{2k−1}⟨⟨a_{1}, …, a_{2k−2}⟩⟩ vanishes since

ν_{2}(2(2k − 2)! S(2k − 1, 2k − 2)c_{2k−1}) = 1 + ν_{2}((2k − 2)!)
+ν_{2}(S(2k − 1, 2k − 2)) + ν_{2}(c_{2k−1})
= 1 + 2k − 2 − d(2k − 2)
+ν_{2}(S(2k − 1, 2k − 2)) + ν_{2}(c_{2k−1})

(by Legendre)

≥ 2k − 1 − d(2k − 2)
+d(2k − 2) − d(2k − 1) + ν_{2}(c_{2k−1})

(by Theorem 2.2)

≥ 2k − 1 − d(2k − 1) + r − (2k − 1) + d(2k − 1)
= r

and 2^{r}W_{t}(F) = 0. Equation (2) is thus equivalent to

(2k − 2)! c_{2k−2}⟨⟨a_{1}, …, a_{2k−2}⟩⟩ = 0

and it follows, since ν_{2}((2k − 2)!) = ν_{2}((2k − 1)!) < r that c_{2k−2} = 2^{r−ν_{2}((2k−2)!)}b_{2k−2} for some b_{2k−2} ∈ ℤ.

Using the same technique and observing that

ν_{2}(2^{q−n}n! S(q, n)c_{q}) = q − n + ν_{2}(n!) + ν_{2}(S(q, p)) + ν_{2}(c_{q})
= q − d(n) + ν_{2}(S(q, n)) + ν_{2}(c_{q})

(by Legendre) ≥ q − d(n) + d(n) − d(q) + r − q + d(q) (by Theorem 2.2)

= r

for all n < q and that ν_{2}(n!) < r for all 1 ≤ n ≤ 2k − 1, the set of equations (1) is equivalent to the set of equations

n! c_{n}⟨⟨a_{1}, …, a_{n}⟩⟩ = 0 , where n = 1, …, 2k − 1.

The solutions are,

c_{n} = 2^{r−ν_{2}(n!)}b_{n} for some b_{n} ∈ ℤ.

We can thus rewrite,

f(X) = c_{2k−1}X^{2k−1} + ... + c_{1}X + c_{0}

as

f(X) = 2^{r−ν_{2}((2k−1)!)}b_{2k−1}X^{2k−1} + 2^{r−ν_{2}((2k−2)!)}b_{2k−2}X^{2k−2} +
… + 2^{r−ν_{2}((2j+1)!)}b_{2j+1}X^{2j+1} + b_{2j}2^{r−ν_{2}((2j)!)}X^{2j} + … + 2^{r−ν_{2}(1!)}b_{1}X + 2^{r−ν_{2}(0!)}b_{0}
= 2^{r−ν_{2}((2k−2)!)}X^{2k−2}(b_{2k−1}X + b_{2k−2}) + … + 2^{r−ν_{2}((2j)!)}X^{2j}(b_{2j+1}X + b_{2j}) + … + 2^{r}(b_{1}X + b_{0})

or equivalently,

f(X) ∈ J_{e,r}

i.e.

A_{t}(F) ⊂ J_{e,r}

and using the other inclusion of the previous lemma

A_{t}(F) = J_{e,r}.

Corollary 3.4. Let F be a nonreal field of finite level s(F) = 2^{r−1} such that s(F)(I(F))^{2k−1} ≠ 0, where k = k(r) is uniquely determined by ν_{2}((2k − 2)!) < r ≤ ν_{2}((2k)!). Then

A(F) = J_{r}.

Proof. Since s(F) = 2^{r−1}, we have that 2^{r}W (F) = 0. Moreover, if F is a nonreal field, I_{t}(F) = I(F). The non-vanishing condition on the power of the fundamental ideal implies that

A_{e}(F) = J_{e,r}.

If ϕ is an odd-dimensional form in W (F), then ϕ ⊥ − ⟨1⟩ is an even-dimensional form in W (F). This implies that

A_{o}(F) = J_{o,r}.

And finally, since J_{e,r} and J_{o,r} are comaximal ideals, we get

A(F) = A_{e}(F) ∩ A_{o}(F) = J_{e,r} ∩ J_{o,r} = J_{e,r} · J_{o,r} = J_{r}.

Corollary 3.5. Let F be a real field of finite height h(F) = 2^{r} such that \frac{1}{2}h(F)(W_{t}(F))^{2k−1} ≠ 0, where k = k(r) is uniquely determined by ν_{2}((2k − 2)!) < r ≤ ν_{2}((2k)!). Then

A_{t}(F) = J_{e,r}.

Proof. By the definition of the height h(F) = 2^{r}, we have that 2^{r}W (F) = 0. Moreover, if F is a real field, I_{t}(F) = W_{t}(F). The non-vanishing condition on the power of the torsion ideal implies that

A_{t}(F) = J_{e,r}.

Remark 3.6. In [5], examples are given of non-real fields F satisfying the conditions of Corollary 3.4. Examples of real fields of arbitrary height h(F) = 2^{r} satisfying the conditions of Corollary 3.5 can be found. These results will be published later.

Remark 3.7. One can show that, for a field F, satisfying 2^{r}W (F) = 0, but not satisfying the non-vanishing condition 2^{r−1}(I_{t}(F))^{2k−1} ≠ 0, the torsion annihilator ideal A_{t}(F) always differs from J_{e,r}. A set of generators for the ideal A_{t}(F) in these cases is, in general, not known.

References

[1] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, a foundation for computer science, (Addison Wesley, 1989).

[2] T. Y. Lam, The algebraic theory of quadratic forms, (Reading Mass., 1973).

[3] D. W. Lewis, Annihilating polynomials for quadratic forms, IJMMS 27, No. 7, 449–455 (2001).

[4] D. W. Lewis, Witt Rings as Integral Rings, Invent. Math. 90, 631–633 (1987).

[5] V. Ongenae, J. Van Geel, Polynomials Annihilating the Witt Ring, Math. Nachr. 185, 213–226 (1997).

[6] J. Riordan, Combinatorial Identities, (Krieger, 1979).

[7] W. Scharlau, Quadratic and Hermitian forms, (Springer-Verlag Berlin-Heidelberg-New York-Tokyo, 1985).

Stefan De Wannemacker Department of Mathematics University College Dublin Belfield, Dublin Ireland

e-mail: sdwannem@maths.ucd.ie