SUMS OF SQUARES IN ALGEBRAIC FUNCTION FIELDS OVER A COMPLETE DISCRETE VALUED FIELD

KARIM JOHANNES BECHER, DAVID GRIMM, AND JAN VAN GEEL

Abstract. A recently found local-global principle for quadratic forms over function fields of curves over a complete discrete valued field is applied to the study of quadratic forms, sums of squares, and related field invariants. Keywords: isotropy, local-global principle, real field, sums of squares, u-invariant, pythagoras number, valuation, algebraic function fields Classification (MSC 2000): 11E04, 11E10, 11E25, 12D15, 12F20, 12J10, 14H05

1. Introduction

Let K be a field of characteristic different from 2 and F/K an algebraic function field (i.e. a finitely generated extension of transcendence degree one). The study of quadratic forms over F is generally difficult, even in such cases where the quadratic form theory over all finite extensions of K is well understood. It can be considered complete in the cases where K is algebraically closed, real closed, or finite, but it is wide open for example when K is a number field.

A breakthrough was obtained recently in the situation where the base field K is a nondyadic local field. Parimala and Suresh [15] proved that in this case any quadratic form of dimension greater than eight over F is isotropic. Harbater, Hartmann, and Krashen [8] obtained the same result as a consequence of a new local–global principle for isotropy of quadratic forms over F. The local conditions are in geometric terms, relative to an arithmetic model for F. A less geometric version of the local–global principle, in terms of the discrete rank one valuations of F, was obtained by Colliot-Thélène, Parimala, and Suresh [4]; see (6.1) below. Both versions of the local–global principle hold more generally when K is complete with respect to a non-dyadic discrete valuation.

In this article we use the local–global principle to study sums of squares in F and to obtain further results on quadratic forms over F. This is of particular interest in the case where K is the field of Laurent series k((t)) over a (formally) real field k. In (6.2) we show that the upper bound on the dimension of anisotropic torsion forms over algebraic function fields over K is the double of the corresponding upper bound for algebraic function fields over k. In the case where k is real closed we show in (6.10) that any sum of squares in F can be expressed as a sum of three squares and further prove the finiteness of ∑ F^{2}/D_{F} (2), the quotient of the group of nonzero sums of squares modulo the subgroup of sums of two squares in F. We further study two conjectures, (4.9) and (4.10), on the behavior of the pythagoras number of a rational function field under extension of the field of constants, and we show in (6.9) that both conjectures are equivalent.

Our methods involve valuation theory, quadratic form theory, and some algebraic geometry. As standard references we refer to [7] for valuation theory and to [10] for quadratic form theory. Results needed from algebraic geometry are cited from [11].

This article grew out of results obtained in the PhD-thesis of D. Grimm under the supervision of K.J. Becher at Universität Konstanz.

Date: May 26, 2012.

2. Valuations

For a ring R we denote by R^{×} its group of invertible elements.

Let K be a field. Given a valuation on K, we denote by 𝒪_{v} the valuation ring of v, by 𝔪_{v} its maximal ideal, and by κ_{v} the residue field, and we call v dyadic if κ_{v} has characteristic 2, nondyadic otherwise. Given a local ring R contained in K, we say that a valuation v of K dominates R if 𝔪_{v} ∩ R is the maximal ideal of R. Given a field extension L/K, we say that a valuation v of L is unramified over K if v(L^{×}) = v(K^{×}).

A valuation with value group ℤ is called a ℤ-valuation. Any discrete valuation of rank one can be identified (via a unique isomorphism of the value groups) with a ℤ-valuation. A commutative ring is the valuation ring of a ℤ-valuation if and only if it is a regular local ring of dimension one (cf. [12, (11.2)]); such rings are called discrete valuation rings.

Lemma 2.1. Let w_{1} and w_{2} be two valuations on K such that 𝔪_{w_{1}} ⊆ 𝒪_{w_{2}}. Then 𝒪_{w_{1}} ⊆ 𝒪_{w_{2}} or 𝒪_{w_{2}} ⊆ 𝒪_{w_{1}}.

Proof. If 𝔪_{w_{1}} ⊆ 𝔪_{w_{2}}, then 𝒪_{w_{1}} ⊇ 𝒪_{w_{2}}, otherwise for any choice of t ∈ 𝔪_{w_{1}} \ 𝔪_{w_{2}} we have t^{−1} ∈ 𝒪_{w_{2}} and 𝒪_{w_{1}} = t^{−1}(t𝒪_{w_{1}}) ⊆ t^{−1}𝔪_{w_{1}} ⊆ 𝒪_{w_{2}}. □

The property for a valuation to be henselian is characterized by a list of equivalent conditions, including the statement of Hensel’s Lemma, hence satisfied in particular by complete valuations; see [7, Sect. 4.1].

Proposition 2.2. Let v be a henselian ℤ-valuation on K. Then v is the unique ℤ-valuation on K.

Proof. By [7, (2.3.2)] for distinct ℤ-valuations w_{1} and w_{2} on K one has 𝒪_{w_{1}} ⊈ 𝒪_{w_{2}} and 𝒪_{w_{2}} ⊈ 𝒪_{w_{1}}. Consider now a ℤ-valuation w on K. Since v is henselian we have 1 + 𝔪_{v} ⊆ K^{×}^{n} for all n ∈ ℕ prime to the characteristic of κ_{v}. As w(K^{×}) = ℤ, this implies that 1 + 𝔪_{v} ⊆ 𝒪_{w}^{×} and thus 𝔪_{v} ⊆ 𝒪_{w}. Now (2.1) yields that 𝒪_{w} = 𝒪_{v}. □

Let X always denote a variable over a given ring or field.

Proposition 2.3. Let R be a local domain with maximal ideal 𝔪 and residue field k. Let p ∈ R[X] be monic and such that \overline{p} ∈ k[X], the reduction of p modulo 𝔪, is irreducible. Then R[X]/(p) is a local domain with maximal ideal (𝔪[X] + (p))/(p) and residue field k[X]/(\overline{p}). The ring R[X]/(p) has the same dimension as R. Moreover, if R is regular, then R[X]/(p) is regular.

Proof. Note that 𝔪[X] + (p) is a maximal ideal of R[X]. Consider a maximal ideal M of R[X] containing p and set 𝔭 = M ∩R. Since R[X]/(p) is an integral extension of R, it follows using [12, (9.3) and (9.4)] that both rings have the same dimension. Moreover, the field R[X]/M is an integral extension of R/𝔭, whereby R/𝔭 is a field. It follows that 𝔭 = 𝔪 and thus M = 𝔪[X] + (p). This shows that 𝔪[X] + (p) is the unique maximal ideal of R[X] containing p. Hence, R[X]/(p) is a local domain with maximal ideal (𝔪[X] + (p))/(p) and residue field k[X]/(\overline{p}). Any set of generators of 𝔪 in R yields a set of generators of (𝔪[X] + (p))/(p) in R[X]/(p). In particular, if R is regular, then so is R[X]/(p). □

Corollary 2.4. Let T be a discrete valuation ring of K with residue field k. Let p ∈ T [X] be monic and such that \overline{p} ∈ k[X] is irreducible. Then T [X]/(p) is a discrete valuation ring with field of fractions K[X]/(p) and residue field k-isomorphic to k[X]/(\overline{p}).

Proof. Since a discrete valuation ring is the same as a regular local ring of dimension one, the statement follows from (2.3). □

We want to mention the following partial generalization of (2.4).

Proposition 2.5. Let T be a valuation ring of K with residue field k and let ℓ/k be a finite field extension. There exists a finite field extension L/K with [L : K] = [ℓ : k] and a valuation v on L dominating T and unramified over K whose residue field is k-isomorphic to ℓ.

Proof. It suffices to consider the case where ℓ = k[x] for some x ∈ ℓ. Let 𝔪 denote the maximal ideal of T. Let p ∈ T [X] be a monic polynomial whose residue \overline{p} in k[X] is the minimal polynomial of x over k. Then p is irreducible in K[X], so L = K[X]/(p) is a field. We obtain from (2.3) that R = T [X]/(p) is a local domain with maximal ideal M = (𝔪[X] + (p))/(p) and residue field k[X]/(\overline{p}). Let v be a valuation on L dominating T. Then T ⊆ R ⊆ 𝒪_{v}, and as M is generated by 𝔪, it follows that v dominates R. Hence, k[X]/(\overline{p}) embeds naturally into κ_{v}. In particular [κ_{v} : k] ≥ deg(\overline{p}) = deg(p) = [L : K]. Using the Fundamental Inequality [7, (3.3.4)] we conclude that v is unramified over K and [κ_{v} : k] = deg(\overline{p}) = [L : K], whereby κ_{v} is k-isomorphic to k[X]/(\overline{p}) and therefore to ℓ. □

3. Valuations on algebraic function fields

In this section we want to relate algebraic function fields over a valued field to algebraic function fields over the corresponding residue field. In particular we show in (3.4) that an algebraic function field over the residue field of a valuation on K can be realized as the residue field of an unramified extension to some algebraic function field over K, and we refine this statement in (3.5) for rational function fields.

In the sequel let T denote a valuation ring, K its field of fractions, and k the residue field of T. (That is, we have T = 𝒪_{v} for a valuation v on K and k = κ_{v}.) We consider the residue fields of valuations dominating T. (The reader may observe that we avoid to speak of extensions of valuations, as this can lead to confusion about the corresponding value groups.) For a field extension F/K and a valuation v on F dominating T, the field k is naturally embedded in the residue field κ_{v}. We often identify residue fields of valuations dominating T up to k-isomorphism, in order to simplify the language.

A finitely generated field extension F/K of transcendence degree one is called an algebraic function field. We say that F/K is algebro-rational if F = L(x) for a finite extension L/K and some element x ∈ F that is transcendental over K.

Proposition 3.1. Let F/K be an algebraic function field and v a valuation on F dominating T. The extension κ_{v}/k is either algebraic or an algebraic function field.

Proof. This is a special case of the Dimension Inequality [7, (3.4.3))]. □

In the sequel x denotes a transcendental element over K. The following gives an improvement of (3.1) for F = K(x).

Theorem 3.2 (Ohm-Nagata). Let v be a valuation on K(x) dominating T. Then κ_{v}/k is either an algebraic or algebro-rational.

Proof. This generalization of [13, Theorem 1] is shown in [14, Theorem]. □

We recall a construction to extend a valuation to a rational function field; in [7, Sect. 2.2] this is called the ‘Gauss extension’.

Proposition 3.3. Let T′ be the localization of T [x] with respect to the prime ideal 𝔪[x] where 𝔪 is the maximal ideal of T. Then T′ is a valuation ring with field of fractions K(x). The residue \overline{x} of x modulo 𝔪[x] is transcendental over k, and the residue field of T′ is k(\overline{x}). The corresponding valuation v on K(x) with 𝒪_{v} = T′, uniquely determined up to equivalence, is unramified over K.

Proof. This follows from [7, (2.2.2)]. □

Proposition 3.4. Let E/k be an algebraic function field. There exists an algebraic function field F/K and a valuation v on F dominating T and unramified over K whose residue field is E.

Proof. We consider the valuation ring T′ given in (3.3) and identify \overline{x} with some element of E transcendental over k. Then E/k(\overline{x}) is a finite extension. By (2.5) there exists a finite field extension F/K(x) with [F : K(x)] = [E : k(\overline{x})] and a valuation v on F dominating T′ and unramified over K(x) with residue field E. Using (3.3) it follows that v is also unramified over K. □

Theorem 3.5. Assume that T ≠ K. Let ℓ/k be a finite separable field extension. There exists a valuation v on K(x) dominating T and unramified over K for which κ_{v}/k is an algebro-rational function field with field of constants ℓ.

Proof. Let α ∈ ℓ be such that ℓ = k(α). Let q ∈ T [Y] be monic and such that the residue \overline{q} in k[Y] is the minimal polynomial of α. Let 𝔪 be the maximal ideal of T. We choose m ∈ 𝔪 \ {0} and set z = m^{−1}q(x) ∈ K(x). Note that z is transcendental over K. Let T′ be the localization of T [z] with respect to 𝔪[z]. Let 𝔪′ be the maximal ideal of T′. By (3.3) T′ is a valuation ring with field of fractions K(z) and residue field k(\overline{z}), and \overline{z} is transcendental over k. Note that \overline{q} remains irreducible in k(\overline{z})[Y].

Consider p = q − q(x) ∈ T′[Y]. As q(x) = mz, taking residues modulo 𝔪′[Y] we have \overline{p} = \overline{q} in k(\overline{z})[Y]. It follows by (2.3) that R = T′[Y]/(p) is a local ring with maximal ideal lying over 𝔪′, with field of fractions K(z)[Y]/(p), and residue field k(\overline{z})[Y]/(\overline{p}). Note that K(z)[Y]/(p) is K(z)-isomorphic to K(x). Using Chevalley’s Theorem [7, (3.1.1)], we obtain a valuation v on K(x) that dominates T′. Then v also dominates T. As p(x) = 0, we have that x is integral over T′, whereby v(X) = 0. We obtain that \overline{q}(\overline{x}) = \overline{p}(\overline{x}) = 0. Hence, \overline{z}, \overline{x} ∈ κ_{v} and \overline{x} is algebraic over k. As \overline{q} is irreducible in k(\overline{z})[Y] we obtain that

[κ_{v} : k(\overline{z})] ≥ [k(\overline{z})[\overline{x}] : k(\overline{z})] = deg(\overline{p}) = deg(p) = [K(x) : K(z)] .

By the Fundamental Inequality [7, (3.3.4)], it follows that v is unramified over K(z) and κ_{v} = k(\overline{z})[\overline{x}] = k[\overline{x}](\overline{z}). Using (3.3) we obtain that v is unramified over K. Since q(\overline{x}) = 0 = \overline{q}(α) and since we consider residue fields up to k-isomorphism, we can identify ℓ = k[α] with k[\overline{x}]. □

Together (3.2) and (3.5) give a full description of the non-algebraic extensions of k that occur as residue fields of valuations on K(x) dominating T.

Assume that the valuation ring T is discrete and consider an algebraic function field F/K. By a regular model for F/T we mean a 2-dimensional integral regular projective flat T -scheme 𝒳 whose function field is K-isomorphic to F. Given a regular model 𝒳 for F/K we denote by 𝒳_{k} its special fiber; by [11, (8.3.3)] 𝒳_{k} is a curve.

Given an integral scheme 𝒳, a point P ∈ 𝒳, and a valuation v on the function field of 𝒳, we say that v is centered at P if v dominates 𝒪_{𝒳,P} , the local ring at P.

Proposition 3.6. Assume that T is a discrete valuation ring. Let F/K be an algebraic function field. Let 𝒳 be a regular model for F/T. Let v be a ℤ-valuation on F dominating T. Then v is centered at a point P of 𝒳 lying in 𝒳_{k}. Moreover, if the extension κ_{v}/k is neither algebraic nor algebro-rational, then 𝒪_{v} = 𝒪_{𝒳,P} where P is the generic point of an irreducible component of 𝒳_{k}.

Proof. By [11, (8.3.17)] v is centered at a point P of the special fiber 𝒳_{k}. Since 𝒳_{k} is a curve, P is either a closed point or the generic point of an irreducible component 𝒳_{k}. In either case 𝒪_{𝒳,P} is a regular local ring.

If P is a closed point of 𝒳_{k}, then by [1, Proposition 3] the extension κ_{v}/k is either algebraic or algebro-rational. Assume now that P is a generic point of 𝒳_{k}.

Then P has codimension one in 𝒳, so 𝒪_{𝒳,P} is a regular local ring of dimension one and thus a discrete valuation ring. As 𝒪_{𝒳,P} is dominated by 𝒪_{v} and both are discrete valuation rings with the same field of fractions, it follows by [7, (2.3.2)] that 𝒪_{v} = 𝒪_{𝒳,P} . □

Proposition 3.7. Assume that T is a complete discrete valuation ring. Let F/K be an algebraic function field. Then there exists a regular model for F/T.

Proof. There exists a regular projective curve over K whose function field is Kisomorphic to F. From this we obtain (e.g. following the first steps in [11, (10.1.8)]) a 2-dimensional projective T -scheme 𝒳 with function field F. Since the structure morphism 𝒳 ⟶ Spec(T) is surjective, by [11, (8.3.1)] it is flat. By [11, (8.2.40)] T is an excellent ring. Since 𝒳 is locally of finite type over T, it follows by [11, (8.2.39)] that 𝒳 is excellent.

Let 𝒳′ ⟶ 𝒳 be the normalization of 𝒳. Since 𝒳 is excellent and projective over T, the normalization 𝒳′ ⟶ 𝒳 is a finite projective birational morphism, by [11, (8.2.39) and (8.3.47)]. The singular locus of 𝒳′ is closed in 𝒳′, by [11, (8.2.38)]. We consider the blowing-up 𝒳′′ ⟶ 𝒳′ along the singular locus of 𝒳′. By [11, (8.1.12) and (8.1.22)] the blowing-up is a birational projective morphism.

We may alternate normalizalization and blowing-up until we reach a scheme that is regular. At each step we obtain a flat projective 2-dimensional T -scheme whose function field is F. By Lipman’s Desingularization Theorem [11, (8.3.44)], after finitely many steps we come to a situation where the T -scheme is regular. □

Corollary 3.8. Assume that T is a complete discrete valuation ring. Let F/K be an algebraic function field. Then there exist only finitely many ℤ-valuations v on F dominating T for which the extension κ_{v}/k is neither algebraic nor algebro-rational.

Proof. By (3.7) there exists a regular model for F/T. The statement follows by applying (3.6) to any such model. □

The result (3.8) can be extended to the situation where T is an arbitrary discrete valuation ring. Moreover, one may ask to characterize the ℤ-valuations on an algebraic function field that dominate a given discrete valuation ring of the base field and for which the residue field extension is neither algebraic nor algebro-rational. We intend to develop these topics in a forthcoming article.

4. Sums of squares and valuations

From now on let K be a field of characteristic different from 2. We denote by ∑ K^{2} the subgroup of nonzero sums of squares in K and, for n ∈ ℕ, by D_{K}(n) the set of nonzero elements that can be written as sums of n squares in K. One calls

s(K) = inf {n ∈ ℕ | −1 ∈ D_{K}(n)} ∈ ℕ ∪ {∞}

the level of K. Recall that K is real if s(K) = ∞ and nonreal otherwise, and in the latter case s(K) is a power of two (cf. [10, Chap. XI, Sect. 2]).

Lemma 4.1. Let v be a valuation on K and n ∈ ℕ. Then s(κ_{v}) ≥ n if and only if v(a_{1}^{2} + ⋯ + a_{n}^{2}) = 2 min{v(a_{1}), …, v(a_{n})} holds for all a_{1}, …, a_{n} ∈ K.

Proof. Both conditions are easily seen to be equivalent to having that any sum of n squares of elements in 𝒪_{v}^{×} lies in 𝒪_{v}^{×}. □

Let Ω(K) denote the set of nondyadic ℤ-valuations on K. For v ∈ Ω(K), let K^{v} denote the corresponding completion of K. For S ⊆ Ω(K) we define a homomorphism

Φ_{S} : K^{×} ⟶ ℤ^{S}, x ⟼ (v(x))_{v∈S} .

If S ⊆ Ω(K) is a finite subset, then it follows from the Approximation Theorem (cf. [7, (2.4.1)] or [11, (9.1.9)]) that Φ_{S} is surjective.

Proposition 4.2. Let S be a finite subset of Ω(K) and n ∈ ℕ. Then

Φ_{S}(D_{K}(n)) = {(e_{v})_{v∈S} ∈ ℤ^{S} | e_{v} ∈ 2ℤ for v ∈ S with s(κ_{v}) ≥ n} .

Proof. For v ∈ Ω(K) with s(κ_{v}) ≥ n we have v(D_{K}(n)) ⊆ 2ℤ by (4.1). This shows that

Φ_{S}(D_{K}(n)) ⊆ {(e_{v})_{v∈S} ∈ ℤ^{S} | e_{v} ∈ 2ℤ for v ∈ S with s(κ_{v}) ≥ n} .

It remains to show the other inclusion. Consider a tuple (e_{v})_{v∈S} ∈ ℤ^{S} such that e_{v} ∈ 2ℤ for all v ∈ S with s(κ_{v}) ≥ n. The aim is to find an element x ∈ D_{K}(n) with Φ_{S}(x) = (e_{v})_{v∈S} . We explain how to obtain such an element, using the Approximation Theorem (cf. [7, (2.4.1)] or [11, (9.1.9)]) several times.

For v ∈ S with e_{v} ∉ 2ℤ, as s(κ_{v}) < n we may choose x_{v,2}, …, x_{v,n} ∈ 𝒪_{v} such that v(1 + x_{v,2}^{2} + ⋯ + x_{v,n}^{2}) > 0. For v ∈ S with e_{v} ∈ 2ℤ we set x_{v,2} = ⋯ = x_{v,n} = 0. For i = 2, …, n we choose x_{i} ∈ K^{×} such that v(x_{i} − x_{v,i}) > 0 for all v ∈ S. We set y = x_{2}^{2} + ⋯ + x_{n}^{2}. For v ∈ S we have v(1 + y) = 0 if e_{v} ∈ 2ℤ and v(1+y) > 0 otherwise. We choose t ∈ K^{×} such that, for all v ∈ S, we have v(t) = 1 if v(1+y) > 1, and v(t) > 1 otherwise. Note that (1+t)^{2}+y ∈ D_{K}(n). For any v ∈ S the value v((1 + t)^{2} + y) is either 0 or 1 and such that v((1 + t)^{2} + y) ≡ e_{v} mod 2ℤ. Choose now z ∈ K^{×} such that 2v(z) = e_{v} − v((1 + t)^{2} + y) for all v ∈ S and set x = z^{2}((1 + t)^{2} + y). Then x ∈ D_{K}(n) and Φ_{S}(x) = (v(x))_{v∈S} = (e_{v})_{v∈S}. □

We say that a valuation v on K is real or nonreal, respectively, if the residue field κ_{v} has the corresponding property.

Corollary 4.3. Let v ∈ Ω(K). If v is real, then v(∑ K^{2}) = 2ℤ, otherwise v(∑ K^{2}) = ℤ.

Proof. This follows from (4.2) applied to S = {v} and all n ∈ ℕ. □

Corollary 4.4. Let n be a positive integer and S a finite subset of Ω(K) such that s(κ_{v}) = 2^{n} for all v ∈ S. Then Φ_{S} induces a surjective homomorphism D_{K}(2^{n+1})/D_{K}(2^{n}) ⟶ (ℤ/2ℤ)^{S}. In particular, |D_{K}(2^{n+1})/D_{K}(2^{n})| ≥ 2^{|S|}.

Proof. By the hypotheses on S and by (4.2), we have Φ_{S}(D_{K}(2^{n+1})) = ℤ^{S} and Φ_{S}(D_{K}(2^{n})) = (2ℤ)^{S}. From this the statement follows. □

The pythagoras number of K is defined as

p(K) = inf {n ∈ ℕ | D_{K}(n) = ∑ K^{2}} ∈ ℕ ∪ {∞} .

Case distinctions in statements involving valuations and pythagoras numbers can often be avoided when p(K) is replaced by s(K) + 1 in case K is nonreal. We therefore set

p′(K) = \{ \begin{matrix}p(K) & if K is real, \\ s(K) + 1 & if K is nonreal. \end{matrix}

Note that for nonreal fields K, we always have s(K) ≤ p(K) ≤ s(K) + 1 = p′(K).

Proposition 4.5. Let v ∈ Ω(K). Then p′(K) ≥ p(K) ≥ p′(κ_{v}). Moreover, if v is henselian, then p′(K) = p(K) = p′(κ_{v}).

Proof. Note that p(K) ≥ p(κ_{v}). If v is real, then κ_{v} and K are real, and we obtain that p′(K) = p(K) ≥ p(κ_{v}) = p′(κ_{v}). Assume that v is nonreal. Applying (4.2) with S = {v} we obtain x_{0}, …, x_{s} ∈ 𝒪_{v}^{×} with s = s(κ_{v}) such that v(x_{0}^{2} + ⋯ + x_{s}^{2}) = 1, and then (4.1) shows that x_{0}^{2} + ⋯ + x_{s}^{2} ∉ D_{K}(s). Hence p(K) ≥ s + 1 = p′(κ_{v}).

Assume finally that v is henselian. Then s(K) = s(κ_{v}), and further p(K) = p(κ_{v}) in case v is real. This yields that p′(K) = p′(k). □

For example, by (4.5) we have that p′(K((t))) = p(K((t))) = p′(K).

Theorem 4.6. Let K be a real field. For n ∈ ℕ the following are equivalent:

(i) p(K(X)) ≤ 2^{n}. (ii) p(L) < 2^{n} for all finite real extensions L/K. (iii) s(L) ≤ 2^{n−1} for all finite nonreal extensions L/K. (iv) p′(L) < 2^{n} for all finite extensions L/K with −1 ∉ L^{×2}.

Proof. See [2, Theorem 3.3] for the equivalence of (i)–(iii); the equivalence of these conditions with (iv) is obvious. □

Corollary 4.7. Let n ∈ ℕ be such that p(K(X)) ≤ 2^{n}. Then p(L(X)) ≤ 2^{n} for any finite field extension L/K.

Proof. If K is nonreal, then p(L(X)) = s(L) + 1 ≤ s(K) + 1 = p(K(X)) ≤ 2^{n}. If K is real and L is nonreal, then s(L) ≤ 2^{n−1} by (4.6) and thus p(L(X)) ≤ 2^{n}. If L is real, then since any finite real extension of L is a finite real extension of K, the equivalence of (i) and (ii) in (4.6) allows us to conclude that p(L(X)) ≤ 2^{n}. □

Theorem 4.8. Let K be henselian with respect to a ℤ-valuation with residue field k. If n ∈ ℕ is such that p(k(X)) ≤ 2^{n}, then p(k(X)) ≤ p(K(X)) ≤ 2^{n}.

Proof. If K is nonreal, then p(K(X)) = s(K) + 1 = s(k) + 1 = p(k(X)), and there remains nothing to show. Assume now that K is real. Then k and k(X) are real. Let v′ denote the ℤ-valuation on K(X) whose valuation ring 𝒪_{v′} is the localization of 𝒪_{v}[X] with respect to the prime ideal 𝔪_{v}[X], as described in (3.3). As κ_{v′} = k(X), we obtain by (4.5) that p(K(X)) ≥ p′(k(X)) ≥ p(k(X)). This shows the first inequality.

Let n ∈ ℕ be such that p(k(X)) ≤ 2^{n}. By (4.6), to prove that p(K(X)) ≤ 2^{n} it suffices to show that p′(L) < 2^{n} for all finite extensions L/K with −1 ∉ L^{×2}. Consider such an extension L/K. Since v is henselian, it extends uniquely to a valuation w on L. This extension is henselian and equivalent to a ℤ-valuation, and its residue field κ_{w} is a finite extension of k. Since w is henselian, we have that p′(L) = p′(κ_{w}) by (4.5) and −1 ∉ κ_{w}^{×2}. Hence, p′(L) = p′(κ_{w}) < 2^{n}, by (4.6). □

The last two statements motivate us to formulate the following two conjectures.

Conjecture 4.9. For any finite field extension L/K, one has p(L(X)) ≤ p(K(X)).

Conjecture 4.10. If K is complete with respect to a nondyadic ℤ-valuation with residue field k, then p(K(X)) = p(k(X)).

We shall see in (6.9) that these two conjectures are equivalent.

5. The u-invariant for algebraic function fields

We refer to [10] for basic facts and terminology from the theory of quadratic forms over fields of characteristic different from two. The u-invariant of K was defined by Elman and Lam [5] as

u(K) = sup {dim(φ) | φ anisotropic torsion form over K} ∈ ℕ ∪ {∞},

where a torsion form is a regular quadratic form that corresponds to a torsion element in the Witt ring.

Proposition 5.1. Let v ∈ Ω(K). Let ψ be a torsion form over κ_{v}. There exist n ∈ ℕ, a_{1}, …, a_{n} ∈ 𝒪_{v}^{×}, and t ∈ K^{×} with v(t) = 1 such that ⟨1, −t⟩ ⊗ ⟨a_{1}, …, a_{n}⟩ is a torsion form over K and such that ψ is Witt equivalent to ⟨\overline{a}_{1}, …, \overline{a}_{n}⟩.

Proof. Assume first that v is nonreal. Then by (4.3) there exists t ∈ ∑ K^{2} with v(t) = 1. For n = dim(ψ) and a_{1}, …, a_{n} ∈ 𝒪_{v}^{×} such that ψ is isometric to ⟨\overline{a}_{1}, …, \overline{a}_{n}⟩, we obtain that ⟨1, −t⟩ ⊗ ⟨a_{1}, …, a_{n}⟩ is a torsion form over K.

Assume now that v is real. Then ψ is Witt equivalent to a sum of binary torsion forms over κ_{v} (cf. [16, Satz 22]). Every binary torsion form over κ_{v} is of the shape ⟨\overline{a}_{1}, \overline{a}_{2}⟩ with a_{1}, a_{2} ∈ 𝒪_{v}^{×} such that −a_{1}a_{2} ∈ ∑ K^{2}. Hence, there exist r ∈ ℕ and a_{1}, …, a_{2r} ∈ 𝒪_{v}^{×} such that ψ is Witt equivalent to ⟨\overline{a}_{1}, …, \overline{a}_{2r}⟩ and −a_{2i−1}a_{2i} ∈ ∑ K^{2} for i = 1, …, r. Then ⟨a_{1}, …, a_{2r}⟩ is torsion form over K. We choose any t ∈ K^{×} with v(t) = 1. Then also ⟨1, −t⟩ ⊗ ⟨a_{1}, …, a_{2r}⟩ is a torsion form over K. □

The following statement was independently obtained in [19, Proposition 5] using different arguments, based on the theory of spaces of orderings.

Proposition 5.2. For v ∈ Ω(K) we have u(K) ≥ 2u(κ_{v}).

Proof. Let v ∈ Ω(K). To prove the statement it suffices to show that to any anisotropic torsion form ψ over κ_{v} there exists an anisotropic torsion form φ over K with dim(φ) ≥ 2 dim(ψ). Let ψ be an anisotropic torsion form over κ_{v}. We choose n ∈ ℕ, a_{1}, …, a_{n} ∈ 𝒪_{v}^{×}, and t ∈ K^{×} with v(t) = 1 as in (5.1). Then ⟨1, −t⟩ ⊗ ⟨a_{1}, …, a_{n}⟩ is a torsion form over K. Let φ denote its anisotropic part. Then φ is a torsion form and isometric to ⟨b_{1}, …, b_{s}⟩ ⊥ −t⟨c_{1}, …, c_{r}⟩ for certain r, s ∈ ℕ and c_{1}, …, c_{r}, b_{1}, …, b_{s} ∈ 𝒪_{v}^{×}. Applying residue homomorphisms (cf. [18, Chap. 6, §2]), it follows that the forms ⟨\overline{b}_{1}, …, \overline{b}_{s}⟩ and ⟨\overline{c}_{1}, …, \overline{c}_{r}⟩ over κ_{v} are Witt equivalent to ψ. As ψ is anisotropic we conclude that dim(φ) ≥ r+s ≥ 2 dim(ψ). □

A generalization of (5.2) for arbitrary nondyadic valuations is given in [3, (5.2)].

Corollary 5.3. Let k be the residue field of a non-dyadic ℤ-valuation on K. For every algebraic function field F/K there exists an algebraic function field E/k such that u(F) ≥ 2u(E).

Proof. Let T denote the discrete valuation ring with field of fractions K and residue field k. Let F/K be an algebraic function field. Choose x ∈ F transcendental over K. Consider the valuation ring T′ in K(x) described in (3.3). Note that T′ is a discrete valuation ring. Since F/K(x) is a finite extension, there exists a ℤ-valuation v on F dominating T′. The residue field E of v is a finite extension of k(\overline{x}), hence an algebraic function field over k. By (5.2) we obtain that u(F) ≥ 2u(E). □

We define

û(K) = \frac{1}{2} sup\{u(F) | F/K algebraic function field \} .

For nonreal fields û coincides with the strong u-invariant defined in [8, Definition 1.2], by the following result.

Corollary 5.4. For any algebraic extension L/K we have

u(L) ≤ \frac{1}{2}u(K(X)) ≤ û(K) .

Proof. If L is a field of odd characteristic p, then the Frobenius homomorphism given by x ⟼ x^{p} shows that any quadratic form over L is obtained by scalar extension from a quadratic form defined over L^{p}. Therefore every torsion form defined over an algebraic extension of K comes from a torsion form defined over a finite separable extension of K. Since any finite separable extension of K is the residue field of a ℤ-valuation v on K(X), the first inequality now follows from (5.2). The second inequality is obvious. □

6. Function fields over complete discrete valued fields

In this section we assume that K is the field of fractions of a complete discrete valuation ring T with residue field k of characteristic different from 2. We want to apply the following reformulation of the local-global principle in [4, (3.1)] to the study of the u-invariant and the pythagoras number of algebraic function fields over K.

Theorem 6.1 (Colliot-Thélène, Parimala, Suresh). Let F be an algebraic function field over K. A regular quadratic form over F of dimension at least 3 is isotropic if and only if it is isotropic over F^{v} for every v ∈ Ω(F).

Proof. This slightly more general version of [4, (3.1)] follows from [9, (9.10)]. □

We now can extend the result [8, Theorem 4.10] to the current setting, thus covering real function fields. C. Scheiderer independently gave a more geometric proof in [19, Theorem 3].

Theorem 6.2. We have û(K) = 2û(k).

Proof. For any algebraic function field E/k, by (3.4) there exists an algebraic function field F/K and a ℤ-valuation on F with residue field E, and using (5.2) we obtain that u(E) ≤ \frac{1}{2}u(F) ≤ û(K). This yields that 2û(k) ≤ û(K).

To prove the converse inequality, we need to show for an arbitrary algebraic function field F/K that u(F) ≤ 4û(k) holds. Fix F/K. By (6.1), any anisotropic form over F remains anisotropic over F^{w} for some w ∈ Ω(F). It thus suffices to show that u(F^{w}) ≤ 4û(k) for every w ∈ Ω(F). Fix w ∈ Ω(F). As u(F^{w}) = 2u(κ_{w}), it suffices to show that u(κ_{w}) ≤ 2û(k). This is clear by the definition of û in case κ_{w}/k is an algebraic function field. If the extension κ_{w}/k is algebraic, it follows from (5.4) that u(κ_{w}) ≤ û(k). In the remaining case, κ_{w} is a finite extension of K. Then κ_{w} is complete with respect to a nondyadic ℤ-valuation, and the corresponding residue field ℓ is a finite extension of k. Then u(κ_{w}) = 2u(ℓ) ≤ 2û(k), by (5.4). □

Corollary 6.3. Let m ∈ ℕ. If u(E) = m for every algebraic function field E/k, then u(F) = 2m for every algebraic function field F/K.

Proof. Let F/K be an algebraic function field over K. Using (6.2) we obtain that u(F) ≤ 2û(K) = 4û(k). By (5.3) there exists an algebraic function field E/k with u(F) ≥ 2u(E). If we assume that u(E) = m holds for every algebraic function field E/k, we obtain that 2û(k) = m and conclude that u(F) = 2m. □

Theorem 6.4. We have u(K(X)) = 2 · sup {u(ℓ(X)) | ℓ/k finite field extension} .

Proof. Let F = K(X). As u(F) ≥ 2, it follows from (6.1) that

u(F) ≤ sup {u(F^{v}) | v ∈ Ω(F)} .

Consider v ∈ Ω(F). We have u(F^{v}) = 2u(κ_{v}). If v is trivial on K, then κ_{v} is a finite extension of K, hence complete with respect to a ℤ-valuation w with 𝒪_{w}∩K = T and whose residue field κ_{w} is a finite extension of k, so that u(κ_{v}) = 2u(κ_{w}) ≤ u(k(X)) by (5.4) and thus u(F^{v}) ≤ 2u(k(X)). If v is nontrivial on K, then by (2.2) and (3.2) κ_{v}/k is either an algebraic extension or algebro-rational. In any case we obtain that u(κ_{v}) ≤ u(ℓ(X)) and thus u(F^{v}) ≤ 2u(ℓ(X)) for a finite extension ℓ/k. This shows that

u(F) ≤ 2 · sup {u(ℓ(X)) | ℓ/k finite field extension} .

Given a finite field extension ℓ/k, it follows from (3.4) that there exists a ℤ-valuation on K(X) with residue field ℓ(X), which by (5.2) implies that u(K(X)) ≥ 2u(ℓ(X)). This shows the claimed equality. □

We turn to the study of sums of squares and the pythagoras number.

Theorem 6.5. Let F/K be an algebraic function field. For any m ≥ 2 we have that D_{F} (m) = F^{×} ∩(⋂_{v∈Ω(K)} D_{F^{v}}(m)). Moreover, p(F) = sup{p′(κ_{v}) | v ∈ Ω(F)}.

Proof. Applying (6.1) to the quadratic forms m × ⟨1⟩ ⊥ ⟨−a⟩ for a ∈ F^{×} shows for any m ≥ 2 the claimed equality of sets. Note that Ω(F) contains a nonreal valuation v, and we have that p(F^{v}) = s(κ_{v}) + 1 ≥ 2. As p(F) ≥ 2, we obtain that

p(F) = inf{m ≥ 2 | D_{F} (m) = D_{F} (m + 1)}
= inf{m ≥ 2 | D_{F^{v}}(m) = D_{F^{v}}(m + 1) for all v ∈ Ω(F)}
= sup{p(F^{v}) | v ∈ Ω(F)} .

Moreover, by (4.5) we have p(F^{v}) = p′(κ_{v}) for every v ∈ Ω(F). □

Theorem 6.6. Let n ∈ ℕ and assume that p(k(X)) ≤ 2^{n} and that ∑ E^{2}/D_{E}(2^{n}) is finite for every algebraic function field E/k. Then p(K(X)) ≤ 2^{n} and ∑ F^{2}/D_{F} (2^{n}) is finite for every algebraic function field F/K.

Proof. By (4.8) we have p(K(X)) ≤ 2^{n}. Consider an algebraic function field F/K. By (6.5) the natural homomorphism

∑ F^{2}/D_{F} (2^{n}) ⟶ ∏_{v∈Ω(F)} ∑ (F^{v})^{2}/D_{F^{v}}(2^{n})

is injective. To prove that ∑ F^{2}/D_{F} (2^{n}) is finite, it thus suffices to show that the set

S = {v ∈ Ω(F) | p(F^{v}) > 2^{n}}

is finite and that ∑ F^{v}^{2}/D_{F^{v}}(2^{n}) is finite for each v ∈ S. Let

Ω_{T} (F) = {v ∈ Ω(F) | 𝒪_{v} ∩ K = T and κ_{v}/k is transcendental }.

Consider v ∈ Ω(F) \ Ω_{T} (F). Then v is trivial on K, so κ_{v} is a finite extension of K. Hence κ_{v} is complete with respect to a ℤ-valuation whose residue field ℓ is a finite extension of k. We conclude that p(F^{v}) = p′(κ_{v}) = p′(ℓ) ≤ 2^{n}. This shows that S ⊆ Ω_{T} (F).

Consider now v ∈ Ω_{T} (F). Then κ_{v}/k is an algebraic function field, in particular | ∑ (F^{v})^{2}/D_{F} (2^{n})| ≤ 2 · | ∑ (κ_{v})^{2}/D_{κ_{v}}(2^{n})|, which is finite by the hypothesis. If κ_{v}/k is algebro-rational, then p′(κ_{v}) ≤ p(K(X)) ≤ 2^{n}, thus p(F^{v}) = p′(κ_{v}) ≤ 2^{n}. The finiteness of S thus follows from (3.8). □

Theorem 6.7. Assume that n ∈ ℕ is such that p(E) ≤ 2^{n} for any algebraic function field E/k. Let F/K be an algebraic function field. Then p(F) ≤ 2^{n} + 1 and the set S = {v ∈ Ω(F) | s(κ_{v}) = 2^{n}} is finite with | ∑ F^{2}/D_{F} (2^{n})| = 2^{|S|}. Moreover, Φ_{S} : F^{×} ⟶ ℤ^{S} induces an isomorphism ∑ F^{2}/D_{F} (2^{n}) ⟶ (ℤ/2ℤ)^{S}.

Proof. For any algebraic extension ℓ/k we have that p′(ℓ) ≤ p(ℓ(X)) ≤ 2^{n}. In particular, we have p′(E) ≤ 2^{n} for any algebro-rational function field E/k. Note further that p′(κ_{v}) = 2^{n} + 1 for any v ∈ S.

Consider v ∈ Ω(F). If v|_{K} is trivial, then κ_{v} is a finite extension of K and therefore complete with respect to a ℤ-valuation w whose residue field κ_{w} is a finite extension of k, whence p′(κ_{v}) = p′(κ_{w}) ≤ 2^{n} by (4.6) and in particular v ∉ S. Suppose that v|_{K} is nontrivial. By (2.2), 𝒪_{v} ∩ K is thus the complete discrete valuation ring on K, and the extension κ_{v}/k is either algebraic or an algebraic function field. If κ_{v}/k is algebraic then p′(κ_{v}) ≤ 2^{n} and in particular v ∉ S. If κ_{v}/k is an algebraic function field, then p(κ_{v}) ≤ 2^{n} and thus p′(κ_{v}) ≤ 2^{n} + 1, with equality holding if and only if v ∈ S, and in this case κ_{v}/k is not algebro-rational. Hence, for v ∈ Ω(F) we have p′(κ_{v}) = 2^{n} + 1 if v ∈ S and p′(κ_{v}) ≤ 2^{n} otherwise.

By (6.5) we conclude that p(F) ≤ p′(F) ≤ 2^{n} + 1 and furthermore

∑ F^{2} = ( ⋂_{v∈S} D_{F^{v}}(2^{n}+1)) ∩ ( ⋂_{v∈S^{c}} D_{F^{v}}(2^{n})) ,

where S^{c} = Ω(F)\S. Moreover, using (3.8) we obtain that S is finite. By (4.4) then Φ_{S} : F^{×} ⟶ ℤ^{S} induces a surjective homomorphism ∑ F^{2}/D_{F} (2^{n}) ⟶ (ℤ/2ℤ)^{S}.

It remains to show that this homomorphism is also injective. In view of (6.5) and the above equality for ∑ F^{2}, it suffices to verify that Φ_{S}^{−1}(2ℤ^{S}) ⊆ ⋂_{v∈S} D_{F^{v}}(2^{n}). Consider x ∈ F^{×} and v ∈ S with v(x) ∈ 2ℤ. Then x = t^{2}y with t ∈ F^{×} and y ∈ 𝒪_{v}^{×} ∩ (∑ F^{2}), so that y + 𝔪_{v} ∈ ∑ κ_{v}^{2}. Since F^{v} is complete and p(κ_{v}) ≤ 2^{n}, it follows that x = t^{2}y ∈ D_{F^{v}}(2^{n}). This shows the claim. □

Theorem 6.8. Let F/K be an algebraic function field. There exists an algebraic function field E/k such that p′(E) ≥ p′(F). Moreover, if F/K is algebro-rational, then one may choose E/k to be algebro-rational.

Proof. If p′(F) ≤ p′(k(X)), we put E = k(X). Now assume that p′(F) > p′(k(X)). Then p(k(X)) < ∞ and thus p(K(X)) < ∞ by (4.8). Since F is K-isomorphic to a finite extension of K, it follows by [17, Chap. 7, (1.13)] that p(F) < ∞. By (6.5) there exists v ∈ Ω(F) such that p(F) = p′(κ_{v}) = p(F^{v}).

Assume first that v|_{K} is trivial. Then κ_{v} is a finite extension of K and thus carries a complete ℤ-valuation w whose residue field κ_{w} is a finite extension of k. We obtain that p′(κ_{v}) = p′(κ_{w}) and thus choose E = κ_{w}(X) to have an algebrorational function field E/k with p′(E) ≥ p′(F).

Assume now that v|_{K} is nontrivial. Then by (2.2) v dominates T. If κ_{v}/k is an algebraic function field, we may choose E = κ_{v} to have that p′(E) ≥ p′(F). By (3.2) if F/K is algebro-rational, then so is E/k. Consider finally the case where κ_{v}/k is an algebraic extension. Since p′(κ_{v}) = p′(F) < ∞, there exists a finite extension ℓ/k contained in κ_{v}/k with p′(ℓ) ≥ p′(κ_{v}), and thus we may choose E = ℓ(X) to have p′(E) ≥ p′(ℓ) ≥ p′(F). □

Corollary 6.9. We have p′(K(X)) = sup {p′(ℓ(X)) | ℓ/k finite field extension}.

Proof. The statement is trivial if k is nonreal. Assume that k is real. Given an arbitrary finite extension ℓ/k, by (3.5) there is a ℤ-valuation on K(X) with residue field ℓ(X), whereby (4.5) yields that p′(ℓ(X)) ≤ p′ (K(X)). On the other hand, by (6.8), there exists a finite extension ℓ/k with p′ (K(X)) ≤ p′(ℓ(X)). □

Note that (6.9) shows the equivalence of the two conjectures (4.9) and (4.10).

Recall that the field K is said to be hereditarily quadratically closed if L^{×} = L^{×2} for every finite field extension L/K. The following result applies in particular to the situation where R is a real closed field.

Theorem 6.10. Let n ∈ ℕ and K = R((t_{1})) … ((t_{n})) for a field R such that R(\sqrt{−1}) is hereditarily quadratically closed. Let F/K be an algebraic function field. Then u(F) = 2^{n+1}, 2 ≤ p(F) ≤ 3, and the group ∑ F^{2}/D_{F} (2) is finite.

Proof. As F is a finite extension of a rational function field, it follows by [10, Chap. VIII, (5.7)] that p′(F) ≥ p(F) ≥ 2. We prove the statement by induction on n. For n = 0 we obtain from [6, Theorem] that u(F) = 2 and conclude by [10, Chap. XI, (6.26)] that p(F) = 2, hence ∑ F^{2} = D_{F} (2) and p′(F) ≤ 3. Assume that n > 0. Applying the induction hypothesis to all algebraic function fields over k = R((t_{1})) … ((t_{n−1})), we obtain by (6.3) that u(F) = 2^{n+1}, by (6.8) that p′(F) ≤ 3, and by (6.6) that ∑ F^{2}/D_{F} (2) is finite. □

Acknowledgments. The authors would like to thank Yong Hu for helpful discussions in the context of (3.6). This work was supported by the Deutsche Forschungsgemeinschaft (project Quadratic Forms and Invariants, BE 2614/3), by the Swiss National Science Foundation (Grant 200020-124785/1), and by the Zukunftskolleg, Universität Konstanz.

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Universität Konstanz, Zukunftskolleg / FB Mathematik und Statistik, 78457 Konstanz, Germany.

E-mail address: becher@maths.ucd.ie

École Polytechnique Fédŕale de Lausanne, EPFL-SB-MATHGEOM-CSAG, Station 8, 1015 Lausanne, Switzerland.

E-mail address: David.Grimm@epfl.ch

Universiteit Gent, Vakgroep Zuivere Wiskunde en Computeralgebra, Galglaan 2, 9000 Gent, Belgium.

E-mail address: jvg@cage.ugent.be