THE MINIMAL HEIGHT OF QUADRATIC FORMS OF GIVEN DIMENSION

JURGEN HURRELBRINK, NIKITA A. KARPENKO, AND ULF REHMANN

In Memory of Martin Kneser

Abstract. Given an arbitrary n, we consider anisotropic quadratic forms of dimension n over all fields of characteristic ≠ 2 and prove that the height of an n-dimensional excellent form (depending on n only) is the (precise) lower bound of the heights of all forms of dimension n.

Contents

1. Introduction 1 2. Relations between heights 4 3. One more property of splitting patterns 5 4. Proof of the Main Theorem 6 5. Final Example 7 References 8

1. Introduction

Excellent quadratic forms and their fundamental properties have been discovered by Knebusch [9] in 1977. They do occur over arbitrary fields. The simplest examples are sums of squares, which are always excellent. This shows that, over suitable fields, anisotropic excellent forms do occur in any dimension.

The definition of excellent forms is closely related to Pfister forms [10]. A quadratic form ϕ is a Pfister neighbor [9], if it is isomorphic to a subform of a quadratic form π such that π is similar to a Pfister form and dim ϕ > (dim π)/2. In this case, the form π and the orthogonal complement of ϕ inside π are uniquely determined (up to an isomorphism) by ϕ. A quadratic form is called excellent, if its dimension is ≤ 1 or if it is a Pfister neighbor with an excellent complement.

If a quadratic form ϕ over a field F is anisotropic and dim ϕ ≥ 2, its first higher Witt index is the Witt index of the form ϕ_{F (ϕ)} = ϕ ⊗_{F} F (ϕ), where F (ϕ) denotes the function field of the quadric defined by ϕ. The anisotropic kernel (ϕ_{F (ϕ)})_{an} is the first higher anisotropic kernel of ϕ (see [10] for the definitions of the usual Witt index and of the usual anisotropic kernel of a quadratic form). Repeating this construction gives, for each ϕ, a sequence of field extensions of F which is called the (generic) splitting tower of ϕ. It is generic in the sense that, for each extension L of F, a specialization from one of the layers K of the splitting tower of ϕ to L can be defined such that the indices of ϕ_{K} and ϕ_{L} are the same.

Date: December 9, 2005.

Key words and phrases. Quadratic forms, Witt indices, Chow groups, correspondences. 2000 Mathematical Subject Classifications: 11E04; 14C25.

The second and third authors were partially supported by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00287, KTAGS. The James D. Wolfensohn Fund and The Ellentuck Fund support is acknowledged by the second author.

The length of the splitting tower of ϕ is called the height 𝔥(ϕ), and the respective anisotropic kernels of ϕ over the field extensions of its splitting tower are its higher anisotropic kernels.

These notions were introduced and extensively studied as well by Knebusch in [8], [9]. It turned out that, for anisotropic excellent forms ϕ of given dimension, the dimensions of the higher kernels are completely independent of the underlying excellent form and even of the ground field F. Knebusch also gave another characterization of excellent forms in [9]: A quadratic form is excellent if and only if all of its higher anisotropic kernels are already defined over F itself. (It should be mentioned here that Knebusch worked over fields of characteristic ≠ 2, but recent research by Hoffmann and Laghribi [1] showed that the same phenomena do occur in characteristic 2.)

In [2], the notion of the splitting pattern of a quadratic form ϕ was defined as the sequence of all possible indices of ϕ over some field extension. From the description above it is clear that all these indices do occur already over the field extensions of the splitting tower of ϕ. In the present note we define the splitting pattern of ϕ as the set of the dimensions of its higher anisotropic kernels, or, equivalently, as the set of the integers dim(ϕ_{E})_{an}, where E runs over all field extension of F.

Every nonnegative integer n is uniquely representable as an alternating sum of 2-powers:

n = 2^{p_{0}} − 2^{p_{1}} + 2^{p_{2}} − ⋯ + (−1)^{r−1}2^{p_{r−1}} + (−1)^{r}2^{p_{r}}  (∗)

with integers p_{0}, p_{1}, …, p_{r} satisfying p_{0} > p_{1} > ⋯ > p_{r−1} > p_{r} + 1 > 0. For an excellent form, it was shown in [2] that its splitting pattern can be determined from (∗) in an easy way (see [2] for details). In particular, the height 𝔥(ϕ) of an anisotropic excellent quadratic form ϕ of dimension n is given by 𝔥(ϕ) = r if n is even and 𝔥(ϕ) = r − 1 if n is odd.

Properties of anisotropic excellent quadratic forms suggest that the excellent forms should be a kind of “object of lowest complexity” among anisotropic forms of a given dimension. In some sense they can be considered as generalizations of Pfister forms to arbitrary dimensions. (Some aspects of their splitting behavior are described in [2, cor. 2.14], which seem to support the conjecture that anisotropic excellent forms provide examples of quadratic forms of minimal canonical dimension in the sense of [6], for quadratic forms of given dimension. An object of “highest complexity” for forms of given dimension clearly is the “generic quadratic form” of that dimension, whose height and canonical dimension are maximal for that dimension.) Hence the authors of [2] conjectured that, for example, the height of an anisotropic excellent form should be the minimal possible height for anisotropic forms of given dimension.

However, a proof of that conjecture so far could not be given with “classical” methods of the theory of quadratic forms. But it was observed by the first and third authors that the results in [4] (see Theorem 1.2 here) could be used to give a simple proof for forms of odd dimension, and, with a slight modification, for all forms of height ≤ 4. In order to give a general proof, the methods of [4] had to be refined, which was done by the second named author in [5], and, in addition, by proving the content of section 3. These ingredients allowed him to assemble a general proof. The combined methods yielded the proof given in this note.

We shall write P (n) for the set {p_{0}, p_{1}, …, p_{r}} occurring in (∗) (note that p_{r} coincides with the 2-adic order v_{2}(n) of n). Let us point out that, for n = 0, our representation is the empty sum, so that P (0) = ∅.

The height 𝔥(n) of the integer n is the number of positive elements in P (n) (so, 𝔥(n) is the number |P (n)| of all elements in the set P (n) for even n, while 𝔥(n) = |P (n)| − 1 for odd n).

Let us state our Main Theorem:

Theorem 1.1. For any anisotropic quadratic form ϕ over a field of characteristic ≠ 2, one has

𝔥(ϕ) ≥ 𝔥(dim ϕ) .

For even dim ϕ, Theorem 1.1 is proved in section 4; for odd dim ϕ, see Corollary 2.4. Sometimes, instead of the splitting pattern {dim(ϕ_{E})_{an}} of ϕ, it is more convenient to

consider the set {𝔦(ϕ_{E})} (with E running over all field extensions of F), where 𝔦(ϕ_{E}) is the Witt index of ϕ_{E}. The elements of this set are called the absolute higher Witt indices of ϕ (the notion of the higher Witt indices has been introduced in [2]). Putting them in ascending order we assign to them the numbers 𝔧_{0} = 𝔦(ϕ) < 𝔧_{1} < ⋯ < 𝔧_{𝔥} = [(dim ϕ)/2] and refer to 𝔧_{i} = 𝔧_{i}(ϕ) as the i-th (absolute) Witt index of ϕ.

Sometimes it is more convenient to consider the relative higher Witt indices defined for i ∈ [1, 𝔥] as 𝔦_{i}(ϕ) = 𝔧_{i}(ϕ) − 𝔧_{i−1}(ϕ) (for convenience, we also set 𝔦_{0}(ϕ) = 𝔧_{0}(ϕ); this is the usual Witt index 𝔦(ϕ)).

Illustration: Let ϕ be an anisotropic excellent form of dimension n = 42. Then 𝔥(ϕ) = 𝔥(n) = 5 and the splitting pattern of ϕ is given by the set {dim(ϕ_{E})_{an}} = {42, 22, 10, 6, 2, 0}. Accordingly the set {j(ϕ_{E})} of absolute Witt indices is given by {0, 10, 16, 18, 20, 21}, and the sequence of relative Witt indices is given by 10, 6, 2, 2, 1.

The splitting pattern {n_{0} > n_{1} > ⋯ > n_{𝔥}} of ϕ is reconstructed from the relative higher Witt indices and the parity of dim ϕ by the formulae n_{i−1} = n_{i} + 2𝔦_{i} (i ∈ [1, 𝔥]), taking into account that n_{𝔥} is 0 or 1 depending on the parity of dim ϕ (note that all n_{i} have the same parity as dim ϕ). Vice versa, 𝔦_{i} = (n_{i−1} − n_{i})/2 for i ∈ [1, 𝔥].

The determination of possible values of splitting patterns is one of the main problems in the modern theory of quadratic forms. It is known that the splitting pattern of any quadratic form satisfies the following restriction:

Theorem 1.2 ([4]). For any integer i ∈ [1, 𝔥], there exists some m such that 2^{m} < n_{i−1}, 𝔦_{i} ∈ [1, 2^{m}], and 𝔦_{i} ≡ n_{i−1} (mod 2^{m}).

Remark 1.3. There is the following interpretation of Theorem 1.2 in terms of the dyadic expansion of n_{i−1}: for an odd n_{i−1}, the integer 𝔦_{i} is a proper binary suffix of n_{i−1}; for an even n_{i−1}, the integer 𝔦_{i} is a proper binary suffix or a proper 2-power divisor of n_{i−1}.

Remark 1.4 ([8]). It is useful to keep in mind that if {n_{0} > n_{1} > ⋯ > n_{𝔥}} with 𝔥 ≥ 1 is the splitting pattern of an anisotropic quadratic form ϕ over a field F, then {n_{1} > n_{2} > ⋯ > n_{𝔥}} (and, consequently, also {n_{i} > n_{i+1} > ⋯ > n_{𝔥}} for any i ≤ 𝔥) is also the splitting pattern of some quadratic form, namely, of the quadratic form (ϕ_{F (ϕ)})_{an}. In particular, it suffices to announce Theorem 1.2 for i = 1 only.

It turns out (see Corollary 2.4), that Theorem 1.1 in odd dimensions is an easy consequence of Theorem 1.2. In even dimensions the situation is more complicated. So, before beginning with the proof of Theorem 1.1 in even dimension, we establish one more property of splitting patterns (see section 3).

Acknowledgments: This article was partially completed when the second author was visiting the University of Western Ontario in London, Canada. He is extremely grateful to the University and personally to Ján Mináč and his wife Leslie for the support, hospitality, and excellent working conditions. The authors would like to acknowledge also valuable suggestions by S. Gorenflo, W. Jarnicki, and A. Laghribi.

2. Relations between heights

In this section, the set {n_{0} > n_{1} > ⋯ > n_{𝔥}} is the splitting pattern of some anisotropic quadratic form ϕ. The main lemma, due to the third author, is:

Lemma 2.1. For an arbitrary integer i ∈ [1, 𝔥], the difference 𝔡(i) = 𝔥(n_{i−1}) − 𝔥(n_{i}) is as follows:

(I) If the dimension of ϕ is odd, then |𝔡(i)| = 1.

(II) If the dimension of ϕ is even, then |𝔡(i)| ≤ 2 and (+2) if 𝔡(i) = 2, then P (n_{i}) ⊂ P (n_{i−1}) and v_{2}(n_{i}) ≥ v_{2}(n_{i−1}) + 2; (+1) if 𝔡(i) = 1, then the difference P (n_{i}) \ P (n_{i−1}) is either empty or consists of one element p, in which case both integers p − 1 and p + 1 are in P (n_{i−1});

(0) if 𝔡(i) = 0, then the difference P (n_{i}) \ P (n_{i−1}) consists of one element p and either p − 1 or p + 1 is in P (n_{i−1});

(-1) if 𝔡(i) = −1, then the difference P (n_{i})\P (n_{i−1}) consists either of two elements p−1 and p+1 for some p ∈ P (n_{i−1}), or the difference consists of one element; (-2) if 𝔡(i) = −2, then the difference P (n_{i})\P (n_{i−1}) consists of two elements (that is, P (n_{i}) ⊃ P (n_{i−1})); moreover, one of these two elements is equal to p + 1 for some p ∈ P (n_{i−1}).

Proof. We write p_{0}, p_{1}, …, p_{r} for the elements of P (n_{i−1}) in descending order. We have n_{i} = n_{i−1} − 2𝔦_{i}. On the other hand, by Theorem 1.2, there exists a nonnegative integer m such that 2^{m} < n_{i−1}, 𝔦_{i} ≡ n_{i−1} (mod 2^{m}), and 1 ≤ 𝔦_{i} ≤ 2^{m}. The condition 2^{m} < n_{i−1} means that m < p_{0}. Let us take the element p_{s} with maximal even s such that m < p_{s}.

If m = p_{s} − 1, then 𝔦_{i} = 2^{p_{s}−1} − 2^{p_{s+1}} + 2^{p_{s+2}} − … and, therefore,

n_{i} = 2^{p_{0}} − 2^{p_{1}} + ⋯ − 2^{p_{s−1}} + 2^{p_{s+1}} − 2^{p_{s+2}} + ⋯ + (−1)^{r−1}2^{p_{r}} .

If s = r and p_{r−1} +1 = p_{r−2}, we get that P (n_{i}) is P (n_{i−1}) without p_{r} and p_{r−2}. Otherwise, P (n_{i}) is P (n_{i−1}) without p_{s}.

Below in this proof, we are assuming that m < p_{s} − 1.

If s = r (and m < p_{r} − 1), then 𝔦_{i} = 2^{m} and n_{i} = n_{i−1} − 2^{m+1}. In the case when m = p_{r} − 2 we get that P (n_{i}) is P (n_{i−1}) with p_{r} transformed to p_{r} − 1. In the case when m < p_{r} − 2 we get that P (n_{i}) is P (n_{i−1}) with m + 1 added.

Below in this proof, we are assuming that s < r.

If p_{s} − 1 > m > p_{s+1}, then 𝔦_{i} = 2^{m} − 2^{p_{s+1}} + 2^{p_{s+2}} − … and, therefore,

n_{i} = 2^{p_{0}} − 2^{p_{1}} + ⋯ − 2^{p_{s−1}} + 2^{p_{s}} − 2^{m+1} + 2^{p_{s+1}} − 2^{p_{s+2}} + ⋯ + (−1)^{r+1}2^{p_{r}} .

This is the right representation of n_{i} (and, therefore, P (n_{i}) is P (n_{i−1}) with m + 1 added).

It remains to consider the case with m ≤ p_{s+1} (while s < r). For this case, let us first assume that s = r − 1. Then 𝔦_{i} = 2^{m} and n_{i} = n_{i−1} − 2^{m+1}. If m < p_{r} − 2, then P (n_{i}) is P (n_{i−1}) with p_{r} +1 and m+1 added. If m = p_{r} −2, then P (n_{i}) is P (n_{i−1}) with p_{r} removed and p_{r} + 1 and p_{r} − 1 added. In the case with m = p_{r} − 1, one has: if p_{r−1} > p_{r} + 2, then P (n_{i}) is P (n_{i−1}) with p_{r} removed and p_{r} + 1 added; if p_{r−1} = p_{r} + 2, then P (n_{i}) is P (n_{i−1}) with p_{r} and p_{r−1} removed while p_{r} + 1 added. Finally, in the case with m = p_{r}, we have: if p_{r−1} = p_{r} + 2, then P (n_{i}) is P (n_{i−1}) without p_{r−1}, otherwise P (n_{i}) is P (n_{i−1}) with p_{r} + 2 added.

We finish the proof considering the case with m ≤ p_{s+1} and s < r − 1. We have: 𝔦_{i} = 2^{p_{s+2}} − 2^{p_{s+3}} + ⋯ + (−1)^{r}2^{p_{r}} and

n_{i} = 2^{p_{0}} − 2^{p_{1}} + ⋯ + 2^{p_{s}} − 2^{p_{s+1}+1} + 2^{p_{s+1}} − 2^{p_{s+2}} + ⋯ + (−1)^{r+1}2^{p_{r}} .

So, if p_{s} > p_{s+1} +1, then P (n_{i}) is P (n_{i−1}) with p_{s+1} +1 added; otherwise P (n_{i}) is P (n_{i−1}) with p_{s} removed. □

Corollary 2.2. For odd-dimensional ϕ, and for any i ∈ [1, 𝔥], one has

𝔥(n_{i−1}) − 𝔥(n_{i}) ≤ 1 .
□

Remark 2.3. In fact, for odd-dimensional ϕ, we have 𝔥(n_{i}) = 𝔥(n_{i−1}) ± 1, which is true as well for an even-dimensional ϕ, if 𝔦_{i} is a proper binary suffix (and not a proper 2-power divisor) of n_{i−1} (see Remark 1.3), but we will not use this fact explicitly during our proof.

Corollary 2.4. For the height 𝔥(ϕ) of an arbitrary anisotropic quadratic form ϕ of odd dimension n, one has 𝔥(ϕ) ≥ 𝔥(n) (that is, Theorem 1.1 holds in odd dimension).

Proof. We have: 𝔥(n_{𝔥}) = 0 (simply because n_{𝔥} = 1) and (by Corollary 2.2) 𝔥(n_{i−1}) − 𝔥(n_{i}) ≤ 1 for every i ∈ [1, 𝔥]. Therefore, 𝔥(n_{0}) ≤ 𝔥. Since ϕ is assumed to be anisotropic, n = n_{0}, and we are done. □

3. One more property of splitting patterns

In this section, {n_{0} > n_{1} > ⋯ > n_{𝔥}} with 𝔥 ≥ 1 is the splitting pattern of an anisotropic quadratic form ϕ of even dimension n = n_{0}. We write X for the projective quadric given by ϕ (see [5] for the definition of the reduced modulo 2 Chow group C̄h of a variety and for the terminology concerning algebraic cycles on X^{2} = X × X).

Proposition 3.1 ([5, th. 3.3 and th. 5.1]). Let α ∈ C̄h(X^{2}) be the minimal cycle containing h^{0} × l_{0}. Assume that α also contains h^{q} × l_{q} for some integer q ∈ [1, [n/2]) and take the minimal q with this property. Let i ∈ [1, 𝔥) be the maximal integer such that 𝔧_{i} ≤ q (then in fact q = 𝔧_{i} by [5, prop. 3.1]). Then v_{2}(𝔦_{i+1}) ≥ v_{2}(𝔦_{1}). Moreover, if v_{2}(𝔦_{2} + ⋯ + 𝔦_{i}) ≥ v_{2}(𝔦_{1}) + 2 (in particular, if i = 1, in which case the sum 𝔦_{2} + ⋯ + 𝔦_{i} is understood to be 0, while by definition v_{2}(0) = ∞), then v_{2}(𝔦_{i+1}) ≤ v_{2}(𝔦_{1}) + 1 so that v_{2}(𝔦_{i+1}) is equal to v_{2}(𝔦_{1}) or v_{2}(𝔦_{1}) + 1 in this case.

A situation in which the condition on α in Proposition 3.1 is always satisfied, is described in:

Theorem 3.2 ([3], for a more elementary proof see [7]). If n_{1} + 𝔦_{1} (= n_{0} − 𝔦_{1}) is not a 2 power, then the minimal cycle on X^{2} containing h^{0} × l_{0} also contains h^{q} × l_{q} for some q ∈ [1, [n/2]).

The main result of the current section, which is the basement of the proof of Theorem 1.1 for even dimension, is:

Theorem 3.3. Assume that v_{2}(n_{i}) ≥ v_{2}(n_{i−1}) + 2 for some i ∈ [1, 𝔥). Then the open interval (i, 𝔥), contains an integer i′ ∈ (i, 𝔥) such that |v_{2}(n_{i′}) − v_{2}(n_{i−1})| ≤ 1.

Proof. It suffices to consider the case of i = 1 (see Remark 1.4). Note that 𝔥 ≥ 2 (otherwise [1, 𝔥) = ∅). We set p = v_{2}(n_{0}). By assumption, we have v_{2}(n_{1}) ≥ p + 2. Therefore v_{2}(𝔦_{1}) = p − 1. Clearly, 𝔦_{1} + n_{1} is not a power of 2; therefore, by Theorem 3.2, the minimal cycle on X^{2} containing h^{0}×l_{0} also contains h^{q}×l_{q} for some q ∈ [1, [n/2]). Let j be the maximal integer satisfying 𝔧_{j} ≤ q. We are going to show that v_{2}(n_{j}) or v_{2}(n_{j+1}) is in [p − 1, p + 1] for this j. Then we can take i′ = j in the first case and i′ = j + 1 in the second case. Note that i′ ≠ 1, 𝔥 (because of v_{2}(n_{1}) ≥ p + 2, while v_{2}(n_{𝔥}) = ∞; we recall that dim ϕ is even in this section).

By the first part of Proposition 3.1, v_{2}(𝔦_{j+1}) ≥ p − 1. Consequently, by Theorem 1.2, v_{2}(n_{j}) ≥ p−1 as well. Since n_{1} = 2(𝔦_{2}+⋯+𝔦_{j})+n_{j}, it follows that v_{2}(𝔦_{2}+⋯+𝔦_{j})+1 ≥ p−1. If v_{2}(𝔦_{2} + ⋯ + 𝔦_{j}) < p + 1, then v_{2}(n_{j}) = v_{2}(𝔦_{2} + ⋯ + 𝔦_{j}) + 1 ∈ [p − 1, p + 1]. So, it remains to consider the case when v_{2}(𝔦_{2} + ⋯ + 𝔦_{j}) ≥ p + 1, where we may apply the second part of Proposition 3.1 as well, stating that v_{2}(𝔦_{j+1}) ∈ {p − 1, p}. Since now v_{2}(n_{j}) ≥ p + 2 while n_{j} = 2𝔦_{j+1} + n_{j+1}, it follows that v_{2}(n_{j+1}) = v_{2}(𝔦_{j+1}) + 1 ∈ {p, p + 1}. □

Corollary 3.4. Under the condition of Theorem 3.3, we set p = v_{2}(n_{i−1}) (note that p ∈ P (n_{i−1}), while min P (n_{i}) ≥ p + 2). Then there exists i′ ∈ (i, 𝔥) such that the set P (n_{i′}) contains an element p′ with |p′ − p| ≤ 1.

Proof. Take i′ as in Theorem 3.3 and set p′ = v_{2}(n_{i′}). □

4. Proof of the Main Theorem

Proof of Theorem 1.1. We only need to prove Theorem 1.1 for even-dimensional forms (see Corollary 2.4). So, let {n_{0} > n_{1} > ⋯ > n_{𝔥}} with 𝔥 ≥ 1 be the splitting pattern of an anisotropic quadratic form ϕ of even dimension n = n_{0}.

Let H be the set {1, 2, …, 𝔥} of 𝔥 = 𝔥(ϕ) elements. For any i ∈ H, we let 𝔡(i) = 𝔥(n_{i−1}) − 𝔥(n_{i}). Let C be the subset of H consisting of all i ∈ H such that 𝔡(i) = 2 (we recall that 𝔡(i) ≤ 2 for any i ∈ H by Lemma 2.1). We prove Theorem 1.1 by constructing a map f : C → H such that 𝔡(j) ≤ 1 − |f^{−1}(j)| for any j ∈ f(C) (in particular, f(C) ⊂ H \ C). Since the subsets f^{−1}(j) ∪ {j}, where j runs over H \ C, are disjoint and cover H, while the average value of 𝔡 on each such subset is ≤ 1, the average value of 𝔡 on H is ≤ 1, and the statement of Theorem 1.1 follows by the same argument as the proof of the odd-dimensional case (see Corollary 2.4).

To define f, let us consider any i ∈ C. By Lemma 2.1, v_{2}(n_{i}) ≥ v_{2}(n_{i−1})+2. Therefore, by Corollary 3.4, there exists i′ ∈ (i, 𝔥) such that the set P (n_{i′}) contains an element p′ satisfying |p′ − p| ≤ 1 for p = v_{2}(n_{i−1}). Taking the minimal i′ with this property, we set f(i) = i′. Also we define g(i) to be the minimal element of P (n_{f(i)}) such that |g(i)−p| ≤ 1.

The map f is constructed. It only remains to check the stated properties of f.

First of all we note that by the very definition of f, for any j ∈ f(C), the difference of sets P (n_{j}) \ P (n_{j−1}) is nonempty (and so, 𝔡(j) ≠ 2 by Item II+2 of Lemma 2.1). Moreover, this difference contains an element p such that {p − 1, p + 1} ⊄ P (n_{j−1}) (and so, 𝔡(j) ≠ 1 by Item II+1 of Lemma 2.1). Therefore, by Lemma 2.1, 𝔡(j) ≤ 0.

Now let j be an element of f(C) with |f^{−1}(j)| ≥ 2. Let i_{1} < i_{2} be two different elements of f^{−1}(j). Note that i_{1} < i_{2} < j and |p_{2} − p_{1}| > 1 (where p_{1} = v_{2}(n_{i_{1}−1}), p_{2} = v_{2}(n_{i_{2}−1})) by definition of f(i_{1}). We are going to show that 𝔡(j) ≤ −1. As we already know, 𝔡(j) ≤ 0. If 𝔡(j) = 0, then by Item II-0 of Lemma 2.1 the difference P (n_{j}) \ P (n_{j−1}) consists of one element p′ and either p′ − 1 or p′ + 1 is in P (n_{j−1}). Since the difference P (n_{j}) \ P (n_{j−1}) consists of one element p′, we have p′ = g(i_{1}) = g(i_{2}), and it follows that {p_{1}, p_{2}} = {p′ − 1, p′ + 1} (in particular, |p_{2} − p_{1}| = 2). Consequently, the set P (n_{j−1}) contains neither p′ − 1 nor p′ + 1, a contradiction.

Now let j be an element of f(C) with |f^{−1}(j)| ≥ 3. Let i_{1}, i_{2}, i_{3} be three different elements of f^{−1}(j). The equalities g(i_{1}) = g(i_{2}) = g(i_{3}) do not hold simultaneously (simply because the three conditions |p_{2} − p_{1}| = 2, |p_{3} − p_{2}| = 2, and |p_{1} − p_{3}| = 2 can not be satisfied simultaneously). On the other hand, the difference P (n_{j}) \ P (n_{j−1}) can have at most two elements. Therefore we may assume that g(i_{1}) = g(i_{2}) and that g(i_{3}) is different from g(i_{1}) = g(i_{2}). We set p′ = g(i_{1}) = g(i_{2}). We are going to show that 𝔡(j) = −2. As we already know, 𝔡(j) ≤ −1. If 𝔡(j) = −1, then by Item II-1 of Lemma 2.1 the difference P (n_{j}) \ P (n_{j−1}) consists of p̃− 1 and p̃+ 1 for some p̃ ∈ P (n_{j−1}). However p′ is neither p̃ − 1 nor p̃ + 1, a contradiction.

We finish the proof by showing that |f^{−1}(j)| is never ≥ 4. Indeed, if |f^{−1}(j)| ≥ 4, then the difference P (n_{j}) \ P (n_{j−1}) contains two elements p′ and p′′ such that none of p′ ± 1 and of p′′ ± 1 is in P (n_{j−1}), contradicting Lemma 2.1. □

5. Final Example

We close with an illustration: Let ϕ be an anisotropic quadratic form of dimension n over a field of characteristic not 2. Above we proved that 𝔥(ϕ) ∈ [𝔥(n), [n/2]]. For n odd one can be more specific. It is a nice exercise to deduce that 𝔥(ϕ) ≡ 𝔥(n) mod 2 for forms of odd dimension n. So,

𝔥(ϕ) ∈ {𝔥(n), 𝔥(n) + 2, …, (n − 1)/2}.

Example: For n = 21 we have 𝔥(n) = 4, hence 𝔥(ϕ) ∈ {4, 6, 8, 10} and, in fact, each of the four values is the height h(ϕ) of some 21-dimensional anisotropic form ϕ, cf. [11].

References

[1] D. W. Hoffmann and A. Laghribi. Quadratic forms and Pfister neighbors in characteristic 2. Trans. Amer. Math. Soc. 356 (2004), no. 10, 4019–4053.

[2] J. Hurrelbrink and U. Rehmann. Splitting patterns of excellent quadratic forms. J. Reine Angew. Math. 444 (1993), 183–192.

[3] O. Izhboldin and A. Vishik. Quadratic forms with absolutely maximal splitting. Contemp. Math. 272 (2000), 103–125.

[4] N. A. Karpenko. On the first Witt index of quadratic forms. Invent. Math. 153 (2003), 455–462.

[5] N. A. Karpenko. A relation between higher Witt indices (revised version). To appear in Amer. Math.

   Soc. Transl. Available on www.math.uni-bielefeld.de/~karpenko

[6] Nikita A. Karpenko: Canonical dimension of orthogonal groups Linear Algebraic Groups and Related Structures (Preprint Server) 144 (July 2004), 4 p.

[7] N. A. Karpenko and A. S. Merkurjev. Rost projectors and Steenrod operations. Doc. Math. 7 (2002), 481–493.

[8] M. Knebusch. Generic splitting of quadratic forms, I. Proc. London Math. Soc. (3) 33 (1976), 65–93.

[9] M. Knebusch. Generic splitting of quadratic forms, II. Proc. London Math. Soc. (3) 34 (1977), 1–31.

[10] W. Scharlau. Quadratic and Hermitian Forms. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985.

[11] A. Vishik. Motives of quadrics with applications to the theory of quadratic forms. Lect. Notes Math. 1835, Springer, Berlin, 2004.

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

 E-mail address: jurgen@math.lsu.edu

Nikita A. Karpenko, Laboratoire de Mathématiques de Lens, Faculté des Sciences Jean Perrin, Université d’Artois, rue Jean Souvraz SP 18, 62307 Lens Cedex, FRANCE

currently: Institute for Advanced Study, 1 Einstein Dr., Princeton, NJ 08540, USA

 Web page: www.math.uni-bielefeld.de/~karpenko
 E-mail address: karpenko@euler.univ-artois.fr

Ulf Rehmann, Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

 E-mail address: rehmann@mathematik.uni-bielefeld.de