Splitting Patterns and Linear Combinations of Two Pfister Forms

By Jürgen Hurrelbrink* at Baton Rouge and Ulf Rehmann at Bielefeld

Dedicated with respect and gratitude to Martin Kneser on the occasion of his 70^{th} birthday

Published in J. Reine Angew. Math. 495 (1998), 163–174

0. Introduction

In section 1 we present a result on the anisotropy of homogeneous polynomials of degree d ≥ 2 over rings of arbitrary characteristic – with an extremely simple proof. The case d = 2 yields a familiar theorem on quadratic forms over fields in the spirit of a theorem by T. A. Springer [17].

It is still an open problem to classify the splitting patterns of regular quadratic forms of height 2 over fields. In section 2 we study linear combinations of two Pfister forms and, in particular, their higher anisotropic kernels of height 2. Related questions have been investigated, some of them very recently, by many authors, based on the originating work of M. Knebusch [10,11], cf. R. Elman and T. Y. Lam [2], R. W. Fitzgerald [3], B. Kahn [8]. In particular, D. Hoffmann has introduced the so-called twisted Pfister forms in [4], which are a special case of linear combinations of two Pfister forms. In 2.7, 2.8, 2.10 we prove excellence criteria and a classification of even-dimensional good forms of height 2 which are linear combinations of two Pfister forms. The resulting list of splitting patterns includes all patterns of excellent and good forms of even dimension and height 2. It seems to be natural to ask if that list is already the complete list of all splitting patterns of even-dimensional anisotropic quadratic forms of height 2. Illustrations are given, based on section 1.

Let e_{0}, e_{1}, …, e_{M}, … denote the cohomological invariants of quadratic forms suggested by J. K. Arason [1] and resulting from the recent work of Voevodsky [18, 14, cf. 9]. Based on sections 1 and 2, we exhibit in section 3, for arbitrary M ∈ ℕ, pairs of anisotropic sums of two Pfister forms q_{1}, q_{2} with different splitting patterns, for which q_{1}, q_{2} lie in the first power only and q_{1} − q_{2} lies in the (M + 1)^{st} power of the fundamental ideal of the Witt ring; so, q_{1}, q_{2} can be separated by their splitting patterns and q_{1} − q_{2} has trivial cohomological invariants e_{0}, e_{1}, …, e_{M}.

Terminology, Notation (Standard reference: [10, 11].)

For a regular quadratic form q over a field k, we let q_{an} denote its anisotropic kernel, and ind q denote its index, i.e., the dimension of a maximal totally isotropic subspace of the underlying vector space of q. A form is called split if dim q_{an} ≤ 1, and it is called hyperbolic if dim q_{an} = 0.

* Partially supported by LEQSF Grant (1995–97)–RD–A–09 and by the SFB 343, University of Bielefeld

As usual, a quadratic form ∑_{i=1}^{n} a_{i}X_{i}^{2} is denoted by ⟨a_{1}, …, a_{n}⟩, and for n-fold Pfister forms we use the notation ⟨⟨a_{1}, … α_{n}⟩⟩ = ⟨1, a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, a_{n}⟩.

By k(q) we denote the function field of its associated quadric given by q = 0 under the assumption that q is not a hyperbolic plane. In case dim q = 2 the field k(q) is just a quadratic extension of k, otherwise it is regular over k. The field k(q) is a generic k-extension over which q becomes isotropic; every field extension with this property is a k-specialization of k(q).

If q is isotropic (and not a hyperbolic plane), then k(q) is a purely transcendental extension of k and vice versa.

To every regular quadratic form q we associate a finite sequence (k_{ν}, q_{ν}) of pairs consisting of a field extension k_{ν} of k and an anisotropic quadratic form q_{ν} over k_{ν}, inductively defined by k_{0} = k, q_{0} = q_{an}, and k_{ν+1} = k_{ν}(q_{ν}), q_{ν+1} = (q_{k_{ν}})_{an} for ν ≥ 0. The height h(q) of q is the smallest number ν such that q_{ν} is split. The sequence k_{ν}, ν = 0, …, h(q) forms a splitting tower of q, the members of the sequence q_{ν}, ν = 0, …, h(q) are called the higher anisotropic kernels of q.

The splitting pattern SP(q) of q is the sequence of indices i_{ν}(q) := ind q_{k_{ν}(q)}, ν = 0, …, h(q). Its members are the so called higher indices of q. These numbers are exactly given by ind q_{L} when L ranges over all field extensions of k, and hence the splitting pattern is a useful invariant of the quadratic form q. It is easily seen that, for every field extension L of k, we have ind q_{L} ≥ i_{ν}(q) if and only if there is a k-specialization k_{ν}(q) → L ∪ ∞.

Therefore, every sequence k_{0} = k, k_{1}, …, k_{h(q)} of field extensions is called a splitting tower of q if k_{ν} is k-specialization equivalent with k_{ν}(q), i.e., k_{ν} and k_{ν}(q) are mutual k-specializations of each other. Then, clearly, i_{ν}(q) = ind q_{k_{ν}} for ν = 0, …, h(q).

We recall that a quadratic form q is called excellent if all its anisotropic kernels q_{ν} are defined over k.

The highest non-trivial anisotropic kernel q_{h(q)−1} of an anisotropic (non-split) form obviously is of height 1. The forms of height 1 are known to be exactly those which are similar to a subform of a Pfister form of codimension ≤ 1. The Pfister form over k_{h(q)−1} associated to q_{h(q)−1} is uniquely determined by q and is called the leading form of q. The form q is called good if its leading form is defined over k.

For example, a four dimensional form with trivial discriminant has splitting pattern (0, 2) and is excellent, representing its own leading form, while a four dimensional form with non-trivial discriminant d ∈ k^{∗} has splitting pattern (0, 1, 2), and its leading form is defined over k by ⟨1, −d⟩, but its first higher kernel is not defined over k, hence such a form is good, but not excellent.

1. Anisotropic homogeneous polynomials over polynomial rings

In this section we give a very elementary proof of an anisotropy criterion for homogeneous polynomials which is, in the context of quadratic forms, well-known (cf. e.g., [2, 4.7, p. 200], [16, ch. 4, 3.3, p. 150 and ch. 6, 2.6, p. 209], [12, ch. 6, 1.4, p. 145]) and is typically deduced from a result of T. A. Springer [17] on quadratic forms over fields with discrete valuations, but which holds in more generality, and whose proof requires only weak assumptions on the underlying ring and does not need any deeper concepts than that of linear independence.

Let R be a commutative ring, let f, g be homogeneous polynomials of arbitrary degree d ≥ 2 in any number of variables with coefficients in R.

More precisely, let

\begin{aligned}f & = & f(X_{1}, …, X_{n}) & = & ∑_{1≤i_{1},...,i_{d}≤n} & a_{i_{1},...,i_{d}} X_{i_{1}}X_{i_{2}} ⋯ X_{i_{d}} & ∈ R[X_{1}, …, X_{n}], \\ g & = & g(Y_{1}, …, Y_{n′}) & = & ∑_{1≤j_{1},...,j_{d}≤n′} & b_{j_{1},...,j_{d}} Y_{j_{1}}Y_{j_{2}} ⋯ Y_{j_{d}} & ∈ R[Y_{1}, …, Y_{n′}], \end{aligned}

where a_{i_{1},...,i_{d}}, b_{j_{1},...,j_{d}} ∈ R.

Such homogeneous polynomials are called forms.

1.1. Definition. A form f is anisotropic over R if f has no non-trivial zero over R; that is, f(x_{1}, …, x_{n}) = 0 for x_{1}, …, x_{n} ∈ R only if x_{1} = … = x_{n} = 0. Otherwise, f is called isotropic over R.

Let T be an indeterminate over R. Then

f + T g := f(X_{1}, …, X_{n}) + T g(Y_{1}, …, Y_{n′})

is a form of degree d in n + n′ variables with coefficients in the polynomial ring R[T], which is contained in the power series ring R[[T]]. If R is an integral domain, then R(T) and R((T)), respectively, denote the ring of rational functions and the Laurent series ring over R in T.

Our goal is to prove the following theorem.

1.2. Theorem. The form f + T g is anisotropic over anyone of the rings R[T], R(T), R[[T]], R((T)) if and only if f and g are anisotropic over R.

Proof. Clearly if f or g is isotropic, then this is true for f + T g. To show the converse, we assume that f + T g is isotropic over R((T)), that is, there are Laurent series

p_{i}(T) = p_{i,r}T^{r} + p_{i,r+1}T^{r+1} + ⋯, 1 ≤ i ≤ n,
q_{j}(T) = q_{j,r}T^{r} + q_{j,r+1}T^{r+1} + ⋯, 1 ≤ j ≤ n′

with r ∈ ℤ, p_{i,ν}, q_{j,ν} ∈ R for ν ≥ r, such that not all p_{i,r}, q_{j,r} are zero and

(i) 0 = ∑_{1≤i_{1},...,i_{d}≤n} a_{i_{1},...,i_{d}}p_{i_{1}}(T) ⋯ p_{i_{d}}(T) + T ∑_{1≤j_{1},...,j_{d}≤n′} b_{j_{1},...,j_{d}}q_{j_{1}}(T) ⋯ q_{j_{d}}(T).

We collect terms of degree dr in (i):

0 = ∑_{1≤i_{1},...,i_{d}≤n} a_{i_{1},...,i_{d}}p_{i_{1},r} … p_{i_{d},r}T^{dr} = f(p_{1,r}, …, p_{n,r})T^{dr}.

If f is anisotropic (otherwise we are done), we obtain p_{1,r} = ⋯ = p_{n,r} = 0 and, using this information, we continue collecting terms of degree dr + 1. The first sum in (i) contributes by monomials having d − 1 factors p_{i,r}, 1 ≤ i ≤ n and hence are zero, as d ≥ 2. The second sum gives

0 = ∑_{1≤j_{1},...,j_{d}≤n′} b_{j_{1},...,j_{d}}q_{j_{1},r} … q_{j_{d},r}T^{dr+1} = g(q_{1,r}, …, q_{n′,r})T^{dr+1}.

Hence if f is anisotropic it turns out that g is isotropic over R, which proves the theorem.

1.3. Remark. The same technique can be used to prove a more general statement for sums of d forms of degree d as follows:

Let f_{0}, …, f_{d−1} be a collection of d forms of degree d over R. Then the form f_{0} + T f_{1} + … + T^{d−1}f_{d−1} is anisotropic over anyone of the rings mentioned in 1.2 if and only if f_{0}, …, f_{d−1} are anisotropic over R.

The analogous statement does not hold anymore for d + 1 forms of degree d.

2. Linear combinations of two Pfister forms

In this section we assume that k is a field of characteristic different from 2. We are interested in quadratic forms which are linear combinations of two Pfister forms σ, τ:

(∗)  q = uσ ⊥ vτ with u, v ∈ k^{∗}, dim τ = 2^{r} ≤ dim σ = 2^{n}.

In order to study these forms we recall a few basic results.

2.1. Remark (cf. [11, p. 2ff]). A quadratic form q of dimension ≥ 2 is a Pfister neighbor if it is similar to a subform of a Pfister form φ and if dim q > dim φ/2, i.e., q ⊥ q′ ≅ aφ for some a ∈ k^{∗} and some form q′ with dim q′ < dim q. The Pfister form φ is uniquely determined by q, and so is the form q′, which is called the complement form of the Pfister neighbor q. If the Pfister neighbor q is anisotropic, its first higher Witt index is dim q − dim φ/2, which is the access of dim q over the greatest 2-power less than dim q, and its first anisotropic kernel is defined by −q′ over k. The fields k(q) and k(φ) are easily seen to be k-specialization equivalent. A quadratic form is excellent if it is split or if it is a Pfister neighbor with excellent complement.

2.2. Lemma [10, p. 75]. Let τ be a Pfister form of degree ≥ 1 and let q be an anisotropic form over k. Then q_{k(τ)} is hyperbolic if and only if τ divides q.

2.3. Lemma [7, 1.13, p. 118]. Let q be an arbitrary anisotropic quadratic form and let n be maximal such that dim q = 2^{n} + ρ with ρ ≥ 1. Then ind q_{k(q)} ≤ ρ. If q is a Pfister neighbor, then equality holds.

2.4. Remark. Let q = uσ ⊥ vτ be as in (∗) above, with σ, τ both anisotropic.

i) q_{an} is similar to a Pfister form if and only if σ and τ are isometric.

ii) q is of height 1 if and only if either q_{an} is similar to a Pfister form or n > r = 0 and q is isotropic (and hence ind q = 1).

iii) Assume q does not split. If 0 < r < n, then the leading form of q is defined over k by τ. If 0 < r < n − 1, then the highest non-trivial anisotropic kernel of q is defined over k by vτ.

Proof. i) If q_{an} is similar to a Pfister form, then q becomes hyperbolic over both of the function fields of σ and τ, hence the same holds for σ and τ. But then, by 2.2, σ and τ divide each other, hence they are isometric. If, conversely, the latter holds, then q ≅ u(σ ⊥ u^{−1}vσ) ≅ u⟨1, u^{−1}v⟩σ.

ii) A form q is of height 1 if and only if it is similar to a subform of a Pfister form of codimension ≤ 1. Hence the even-dimensional case is obvious, and we may assume that dim q is odd. In that case we have n > r = 0, hence τ = ⟨1⟩. If q is isotropic, then q_{an} is a codimension 1 subform of uσ, hence of height 1. If q is anisotropic, then ind q_{k(q)} = 1 by 2.3, hence it is of height (at least) 2.

iii) This follows directly from [11, Th. 9.6, 9.7, p. 22f.].

Part i) of the following lemma is a generalization of the linkage results in [2, §4, p. 197ff.] and is essentially well-known (cf. [11, 8.11, p. 17] and [4, 3.2, p. 72]).

2.5. Lemma. Let q = uσ ⊥ vτ with u, v ∈ k^{∗} and two anisotropic Pfister forms σ, τ over k.

i) Assume that q is isotropic. Then there is some a ∈ k^{∗} represented by σ such that q ≅ ua(σ ⊥ −τ) and −ua/v is represented by τ. Moreover, ind q is the dimension of a Pfister form of maximal degree dividing both of the forms σ and τ.

ii) If q is anisotropic and α is a Pfister form dividing both forms σ and τ, then ind q_{k(q)} ≥ dim α.

Proof. i) As q is isotropic, there are elements a, b ∈ k^{∗} represented by σ, τ, respectively, with ua + vb = 0. Then we have uσ ⊥ vτ ≅ uaσ ⊥ vbτ ≅ ua(σ ⊥ −τ).

Hence we may assume that q = σ ⊥ −τ. We write σ = ασ_{1}, τ = ατ_{1} with Pfister forms α, σ_{1}, τ_{1}. Then σ ⊥ −τ = α ⊥ −α ⊥ ασ′_{1} ⊥ −ατ′_{1}, where σ′_{1}, τ′_{1} are the pure parts of σ_{1}, τ_{1}. Hence ind q ≥ dim α.

If ind q > dim α, then there is a scalar c ∈ k^{∗} represented by both of ασ′_{1} and ατ′_{1}. Hence α ⊥ ⟨c⟩ is an orthogonal summand of σ and τ and a neighbor of the Pfister form α⟨1, c⟩, hence, by 2.1, the function field of α⟨1, c⟩ splits σ and τ. From 2.2 it then follows that α⟨1, c⟩ divides both forms. Choosing α as a common Pfister divisor of σ, τ of maximal dimension therefore yields ind q = dim α.

ii) Since q_{k(q)} is isotropic, the claim follows immediately from i).

2.6. Corollary. Let σ, τ be two anisotropic Pfister forms of degree n and r, respectively, with n ≥ r ≥ 0, and let q = uσ ⊥ vτ for some u, v ∈ k^{∗} with dim q ≥ 3. i) The splitting pattern of q is a subsequence of

(0, 1, 2^{n−1})  if r = 0
(0, 2^{0}, 2^{1}, …, 2^{r}, 2^{n−1}, 2^{n−1} + 2^{r−1}) if 0 < r ≤ n − 2
(0, 2^{0}, 2^{1}, …, 2^{n−1}, 2^{n−1} + 2^{n−2})  if r = n − 1
(0, 2^{0}, 2^{1}, …, 2^{n})  if r = n.

ii) Let X_{1}, …, X_{n}, Y_{1}, …, Y_{r}, Z be indeterminates over k and let K = k(X_{i}, Y_{j}, Z). Then the splitting pattern of the quadratic K-form

q = ⟨⟨X_{1}, …, X_{n}⟩⟩ ⊥ Z⟨⟨Y_{1}, …, Y_{r}⟩⟩

is precisely given by the corresponding tuple in i).

Proof. i) If K/k is any field extension, then either both of σ, τ are anisotropic, hence ind q = 0 or 2^{i} with 0 ≤ i ≤ r by 2.5, or at least one of σ, τ is isotropic, hence split, and then ind q ∈ {2^{r−1}, 2^{n−1}, 2^{n−1} + 2^{r−1}}. This proves i).

ii) A repeated application of 1.2 for d = 2 shows that σ := ⟨⟨X_{1}, …, X_{n}⟩⟩ and τ := ⟨⟨Y_{1}, …, Y_{r}⟩⟩ are anisotropic; another application of 1.2 then gives the anisotropy of q = σ ⊥ Zτ. For i ∈ {0, …, r}, let L_{i} be the field extension of K obtained by adjunction of the square roots of −Z and X_{ν}/Y_{ν} for ν = 1, …, i. Then q_{L} ≅ σ_{L} ⊥ −τ_{L} = ⟨⟨X_{1}, …, X_{i}⟩⟩(⟨⟨X_{i+1}, …, X_{n}⟩⟩ ⊥ −⟨⟨Y_{i+1}, …, Y_{r}⟩⟩), which is of index 2^{i} by 1.2 and 2.5.

2.7. Lemma. Let q = uσ ⊥ vτ be as in 2.6.

i) We have dim q_{an} = 2^{n} + 2^{r} − 2 ind q. Thus dim q_{an} is a power of 2 if and only if ind q = 2^{i} with i = r − 1 or n − 1 or ind q = 0 and r = n.

ii) 2^{n−1} occurs in SP(q) if and only if r < n or σ ≠ τ.

iii) 2^{r−1} occurs in SP(q) if and only if τ does not divide σ.

iv) If an entry 2^{i} with 0 ≤ i ≤ min {r − 1, n − 2} occurs in SP(q), then the entries 2^{r−1}, 2^{r}, 2^{n−1} occur as well, and q is not excellent.

v) If no entry 2^{i} with 0 ≤ i ≤ r − 1 occurs in SP(q), then q is excellent. The splitting pattern is a suffix of the sequence

(0, 1, 2^{n−1})  if r = 0
(0, 2^{r}, 2^{n−1}, 2^{n−1} + 2^{r−1}) if 0 < r ≤ n − 2
(0, 2^{n−1}, 2^{n−1} + 2^{n−2})  if r = n − 1
(0, 2^{n})  if r = n.

Proof. i) Clear.

ii). We have (q_{k(σ)})_{an} ≅ (vτ_{k(σ)})_{an}, which yields an index 2^{n−1} if and only if τ_{k(σ)} is not hyperbolic. By 2.2, this is true if and only if σ does not divide τ. iii). Similarly, (q_{k(τ)})_{an} ≅ (uσ_{k(τ)})_{an}, yielding an index 2^{r−1} if and only if σ_{k(τ)} is not hyperbolic. By 2.2, this is true if and only if τ does not divide σ.

iv) If an entry 2^{i} with 0 ≤ i ≤ r − 1 occurs in SP(q), then τ does not divide σ because of 2.5 i), hence, by iii), 2^{r−1} occurs in SP(q). If r ≤ n−1, then, by ii), also 2^{n−1} occurs. In order to show that 2^{r} occurs in SP(q) we may assume that r ≤ n − 2. The higher kernel q′ belonging to the higher index 2^{r−1} has dimension 2^{n}. If 2^{r} did not occur in SP(q), then q′ would be of height 2, and since, by 2.4 iii), its highest anisotropic kernel is defined over k, it would be excellent, contradicting the fact that an excellent form of 2-power dimension is of height 1. Hence 2^{r} must occur as well in SP(q).

We have to show that q is not excellent. If r ≤ n − 1, the higher kernel q′ belonging to the higher index 2^{r−1} is of 2-power dimension and of height > 1, hence q is not excellent. If r = n, let i ≤ n − 2 be maximal such that 2^{i} occurs in SP(q). Then the corresponding anisotropic kernel q′ is of dimension 2^{n} + 2^{n} − 2^{i+1}. If q were excellent, the index of q′ would be 2^{n} −2^{i+1}, by 2.3. Hence the entry of SP(q) following 2^{i} would be 2^{i} + 2^{n} − 2^{i+1} = 2^{n} − 2^{i}, which yields a contradiction.

v) By iii), τ divides σ. In case r = n, the forms σ and τ must be equal, and hence q is similar to a Pfister form. If r < n, there is a Pfister form σ_{1} such that q = uτσ_{1} ⊥ vτ = uτ(σ_{1} ⊥ ⟨u^{−1}v⟩), which is a Pfister neighbor of τσ_{1}⟨⟨u^{−1}v⟩⟩. Its complement is the first higher kernel of q, hence q is excellent in case r = n − 1. In case r ≤ n − 2 we know from 2.4 iii) that also the last non-trivial higher kernel is defined over k, hence q is excellent in that case as well. The rest is clear by 2.6 i).

The following theorem describes criteria for excellence of linear combinations of two anisotropic Pfister forms in terms of splitting patterns. It turns out that all splitting patterns of excellent even-dimensional forms of height 2 are already obtained by these forms. Surprisingly, also forms of height 3 show up among the excellent forms.

2.8. Theorem. Let q = uσ ⊥ vτ be a linear combination of two anisotropic Pfister forms σ, τ with u, v ∈ k^{∗}, dim q ≥ 3, and 2^{n} = dim σ ≥ dim τ = 2^{r}.

i) If r = 0, then q is excellent.

ii) Assume 0 < r ≤ n − 2. Then the last three entries of the splitting pattern of q are given by 2^{r}, 2^{n−1}, 2^{n−1} + 2^{r−1}. The form q is excellent if and only if either q is anisotropic and h(q) = 3 or q is isotropic and h(q) = 2.

iii) Assume 0 < r = n − 1. Then the last two entries of the splitting pattern of q are given by 2^{n−1}, 2^{n−1} + 2^{n−2}. The form q is excellent if and only if either q is anisotropic and h(q) = 2 or q is isotropic and h(q) = 1.

iv) Assume r = n. If q is not excellent, then the last two entries of the splitting pattern of q are given by 2^{n−1}, 2^{n}. The form q is excellent if and only if h(q) ≤ 1.

If r < n, then 2^{r−1} appears in the splitting pattern of q if and only if q is not excellent.

Proof. The proof of this theorem is easily obtained from 2.7.

Remark. We recall that, by a result of R. W. Fitzgerald [3, 1.2, p. 340], the odd dimensional good forms are precisely the odd dimensional excellent forms and are always similar to a form of type φ′σ ⊥ ⟨1⟩ with an n-fold Pfister form σ and an l-fold Pfister form φ with pure part φ′. The forms in 2.8 i) are just the particular case l = 1.

2.9. Illustration. For example, for r = 1 and n ≥ 3, all splitting patterns of forms q as in 2.8 are given by

(0, 2, 2^{n−1}, 2^{n−1} + 1), (0, 1, 2, 2^{n−1}, 2^{n−1} + 1), (1, 2, 2^{n−1}, 2^{n−1} + 1), (2, 2^{n−1}, 2^{n−1} + 1)

and they can be realized by the quadratic forms

2^{n} × ⟨1⟩ ⊥ ⟨⟨1⟩⟩, 2^{n} × ⟨1⟩ ⊥ ⟨⟨T ⟩⟩, 2^{n} × ⟨1⟩ ⊥ −⟨⟨T ⟩⟩, 2^{n} × ⟨1⟩ ⊥ −⟨⟨1⟩⟩,

respectively, over k = R(T) with a field R of level greater than 2^{n} in view of 1.2.

The following theorem describes all non-excellent good forms of height 2, which are linear combinations of two Pfister forms q. By 2.4 iii) and by 2.8 i) we know that – with the notation of (∗) – for r = 0 or r < n − 1 the highest anisotropic kernel of q is defined over k. Hence non-excellent forms of height 2 may occur only when r ≥ max {1, n − 1}.

2.10. Theorem. Let q be the anisotropic part of a linear combination of two Pfister forms. The following statements are equivalent.

i) q is of even dimension and of height 2, and its leading form, but not its first higher anisotropic kernel, is defined over k.

ii) q = uσ ⊥ vτ is anisotropic with u, v ∈ k^{∗} and (n − 1)-fold non-isometric Pfister forms σ, τ, which have an (n − 2)-fold common Pfister divisor.

iii) q is similar to (σ_{1} ⊥ −τ_{1})_{an}, with an anisotropic n-fold Pfister form σ_{1} and an anisotropic (n − 1)-fold Pfister form τ_{1} not dividing σ_{1}, which have an (n − 2)-fold common Pfister divisor.

Proof. ii) ⇒ iii). By the assumptions in ii), we have an (n − 2)-fold Pfister form α and elements x, y ∈ k^{∗} such that σ = α⟨⟨x⟩⟩, τ = α⟨⟨y⟩⟩ and hence q = uα⟨⟨x⟩⟩ ⊥ vα⟨⟨y⟩⟩. Since we want to characterize q up to similarity, we may assume x = v, that is, q = α(u⟨⟨v⟩⟩ ⊥ v⟨⟨y⟩⟩) = α⟨u, v, uv, vy⟩ ≅ α(⟨⟨u, v⟩⟩ ⊥ −⟨⟨−vy⟩⟩)_{an}. Since α is a common Pfister divisor of σ, τ of maximal dimension, v, y represent different square classes in k^{∗}. Hence ⟨⟨−vy⟩⟩ is anisotropic, and therefore, 2.5 i) yields that α is a common Pfister divisor of σ_{1} := α⟨⟨u, v⟩⟩ and τ_{1} := α⟨⟨−vy⟩⟩ of maximal dimension. This proves iii).

iii) ⇒ ii). We have an (n − 2)-fold Pfister form α and elements u, v, c ∈ k^{∗} such that q = (σ_{1} ⊥ −τ_{1})_{an} = α(⟨⟨u, v⟩⟩ ⊥ −⟨⟨c⟩⟩)_{an} = α⟨u, v, uv, −c⟩ = uα⟨⟨v⟩⟩ ⊥ vα⟨⟨−c/v⟩⟩. By 2.8 iii), h(q) = 2, hence α is a common Pfister divisor of maximal dimension of σ := α⟨⟨v⟩⟩ and τ := α⟨⟨−c/v⟩⟩, and ii) follows.

ii) ⇒ i) Let q be as in ii). It follows from 2.4, iii) that the leading form of q is defined over q. From 2.6 we deduce that dim q = 2^{n} + 2^{n−1} − 2 · 2^{n−2} = 2^{n}. By the equivalence of ii) and iii) we have h(q) = 2. Since excellent forms of 2-power dimension are of height 1, q is not excellent, i.e., its first higher kernel is not defined over k.

i) ⇒ ii) or iii). Let τ_{1} denote the leading form of q. Then q_{k(τ_{1})} is anisotropic and of height 1, hence a Pfister form and therefore of 2-power dimension, say, dim q = 2^{n}. If q = (uσ ⊥ vτ)_{an}, then, by 2.6, the only possibilities for the index of q and the dimensions of σ, τ being compatible with h(q) = 2 are ind q = 2^{n−2}, dim σ = 2^{n}, dim τ = 2^{n−1}, or ind q = 0, dim σ = 2^{n−1}, dim τ = 2^{n−1}. In the first case, 2.5 gives us an (n − 2)-fold common Pfister divisor of σ, τ, which yields iii). In the second case, the condition h(q) = 2 yields dim (q_{k(q)})_{an} = 2^{n−1}, hence ind q_{k(q)} = 2^{n−1}. By [7, 1.12, p. 118] we obtain that σ_{k(q)} and τ_{k(q)} are anisotropic, hence 2.5 gives us an (n − 1)-fold common Pfister divisor of σ, τ, which yields ii).

2.11. Remark. i) One is interested in the list of splitting patterns of all evendimensional anisotropic quadratic forms of height 2. The higher anisotropic kernels of height 2 of the generic linear combination of two Pfister forms in 2.6 provide us with the following list:

In case 1 ≤ r ≤ n−2, the splitting patterns of the higher kernel forms of dimension 2^{n} − 2^{r}, n ≥ 3, are given by (0, 2^{n−1} − 2^{r}, 2^{n−1} − 2^{r−1}). Thus, by [11, 10.1, p. 28] or [6, 2.8, p. 187], we obtain the complete list of splitting patterns of the excellent height 2 forms.

In case 1 ≤ r = n−1, the splitting patterns of the higher kernel forms of dimension 2^{n}, n ≥ 2, are given by (0, 2^{n−2}, 2^{n−1}). Thus, by [5, p. 11] or [7, 3.9, 3.10, p. 124ff] we obtain the complete list of splitting patterns of the non-excellent good height 2 forms.

In case r = n, the splitting patterns of the higher kernel forms of dimension 2^{n+1} − 2^{n−1} = 3 · 2^{n−1}, n ≥ 2, are given by (0, 2^{n−2}, 2^{n} − 2^{n−2}). They include the splitting patterns of non-good forms obtained by multiplying an Albert form with splitting pattern (0, 1, 3) by a generic Pfister form.

ii) Assume now that q = uσ ⊥ vτ is as in (∗), but that its leading form is not defined over k. It then follows from 2.4 iii) that dim σ = dim τ = 2^{n}, and we obtain from 2.10 that one of the higher anisotropic kernels of q must be of dimension 2^{n+1} − 2^{i+1} with i ∈ {0, …, n − 2}.

If we take i to be maximal with this property, the corresponding anisotropic kernel gives a form of height 2 and of dimension 2^{n+1} − 2^{i+1} with splitting pattern (0, 2^{n−1} − 2^{i}, 2^{n} − 2^{i}).

Replacing the base field by the appropriate generic splitting field we may assume that ind q = 2^{i+1}, and that h(q) = 2. By 2.5 iii), we may write q = ασ_{1} ⊥ −ατ_{1} with some (n − 2)-fold Pfister form α. Hence we obtain q_{an} = α(σ′_{1} ⊥ −τ′_{1}) with σ′_{1}, τ′_{1}, respectively, the pure parts of the 2-fold Pfister forms σ_{1}, τ_{1}.

Hence, in case i = n − 2, the form q_{an} is a product of an Albert form σ′_{1} ⊥ −τ′_{1} and some Pfister form.

If we had i < n − 2, then we would, by a similar reasoning, obtain a form of height 2 of anisotropic dimension 2^{n+1} −2^{i+1}. The smallest possibility for that is n = 3, i = 0, which would give dim q_{an} = 14 and splitting pattern (0, 3, 7). We do not know if such a form exists.

More generally, the question is as follows. We know, by the generic examples of 2.6 ii), that there exist forms of type (σ ⊥ −τ)_{an} with splitting pattern

(0, 1, 3, …, 2^{i} − 1, …, 2^{n} − 1).

So, it would be interesting to know what type of gaps can occur in splitting patterns of forms of the above type, in order to determine whether there exist height 2 splitting patterns of even-dimensional forms not listed in i).

3. Splitting patterns and higher invariants

Voevodsky’s announced confirmation (cf. [18, 14, 9]) of the Milnor Conjecture [13] over fields k of characteristic ≠ 2 provides us with higher invariants of quadratic forms.

Let I = I(k) denote the fundamental ideal of the Witt ring W = W (k) of k. For r = 0, 1, 2, …, the ideal I^{r} is generated by r-fold Pfister forms ⟨⟨a_{1}, …, a_{r}⟩⟩ a_{i} ∈ k^{∗}. Let K_{r}^{M}(k) denote the r-th Milnor K-group of k; it is generated by r-fold symbols (b_{1}, …, b_{r}), b_{i} ∈ k^{∗}. We recall:

3.1. Theorem ([14], cf. [9, p. 5]). For every field k with char k ≠ 2 and every r = 0, 1, 2, …, there is an isomorphism e_{r} : I^{r}/I^{r+1} → K_{r}^{M}(k)/2 with e_{r} induced by ⟨⟨a_{1}, …, a_{r}⟩⟩ → (−a_{1}, …, −a_{r}).

In particular, e_{0}(q) ∈ ℤ/2 is the dimension of q modulo 2, e_{1}(q) ∈ k^{∗}/k^{∗2} is the discriminant of q, and e_{2}(q) ∈ Br_{2}(k) is the Clifford invariant of q.

We are going to exhibit anisotropic quadratic forms q, q′ ∈ I \ I^{2} whose difference lies in a high power of the fundamental ideal, which however can be separated by their splitting patterns.

3.2. Proposition. For every M ∈ ℕ, there exist anisotropic quadratic forms q, q′ over a field k with q, q′ ∈ I \ I^{2} and q − q′ ∈ I^{M+1}, yet SP(q) ≠ SP(q′).

Proof. Given M ∈ ℕ, let F be a field of level s(F) > 2^{M+1}. Set k = F (T) with an indeterminate T. For M = 1 consider

q = ⟨1, 1, 1, 1, 1, T ⟩ = ⟨⟨1, 1⟩⟩ ⊥ ⟨⟨T ⟩⟩ and q′ = ⟨1, 1, 1, T, T, T ⟩ = ⟨⟨1, T ⟩⟩ ⊥ ⟨⟨T ⟩⟩.

By 1.2, q and q′ are anisotropic over k since F is of level > 4. The forms q and q′ are both of dimension 6 and of discriminant −T. So, they lie in I \ I^{2}. Their difference is given by q − q′ = ⟨1, 1, −T, −T ⟩ = ⟨⟨1, −T ⟩⟩ ∈ I^{2}.

The splitting patterns of anisotropic forms of discriminant ≠ 1 in dimension 6 are given by (0, 1, 2, 3) and (0, 2, 3) by 2.6 i) and 2.7 iii). Over k(\sqrt{−T}), the form q splits exactly one hyperbolic plane, so SP(q) = (0, 1, 2, 3). The form q′ is excellent, thus SP(q′) = (0, 2, 3).

For M ≥ 2 consider

q = 2^{M+1} × ⟨1⟩ ⊥ ⟨1, T ⟩ = ⟨⟨1, …, 1⟩⟩ ⊥ ⟨⟨T ⟩⟩

and

q′ = 2^{M} × ⟨1⟩ ⊥ 2^{M} × ⟨T ⟩ ⊥ ⟨1, T ⟩ = ⟨⟨1, …, 1, T ⟩⟩ ⊥ ⟨⟨T ⟩⟩;

both forms q and q′ are sums of an (M + 1)-fold and a 1-fold Pfister form.

By 1.2, q and q′ are anisotropic over k and both lie in I \ I^{2} with difference q − q′ = ⟨⟨1, …, 1, −T ⟩⟩ ∈ I^{M+1}.

By 2.8 with n = M + 1 ≥ 3, r = 1, we know that q has splitting pattern (0, 1, 2, 2^{M}, 2^{M} + 1). The form q′ is excellent of dimension 2^{M+1} + 2; thus SP(q′) = (0, 2, 2^{M}, 2^{M} + 1).

In [15, 3.4.1, p. 251] an example is given of two non-isometric, but similar, quadratic forms having the same Stiefel-Whitney classes. We add:

3.3. Corollary. For every M ∈ ℕ, there exist anisotropic quadratic forms q, q′ ∈ I \ I^{2} over a field k for which q − q′ has trivial invariants e_{0}, e_{1}, …, e_{M}, yet q and q′ are not similar.

Proof. Given M ∈ ℕ, the forms q, q′ from Prop. 3.2 with e_{0}(q − q′) = e_{1}(q − q′), ⋯, e_{M}(q − q′) all being trivial, are not similar, since similar forms have the same splitting pattern.

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Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA

jurgen@math.lsu.edu Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany rehmann@mathematik.uni-bielefeld.de