u-Invariants for forms of higher degree

Both a general and a diagonal u-invariant for forms of higher degree are defined, generalizing the u-invariant of quadratic forms. We give a survey of both old and new results on these u-invariants.     

The automorphism group of certain higher degree forms

We consider symmetric indecomposable d-linear (d>2) spaces of dimension n over an algebraically closed field k of characteristic 0, whose center (the analog of the space of symmetric matrices of a bilinear form) is cyclic, as introduced by Reichstein [B. Reichstein, On Waring’s problem for cubic forms, Linear Algebra Appl. 160 (1992) 1-61]. The automorphism group of these spaces is determined through the action on the center and through the determination of the Lie algebra. Furthermore, we relate the Lie algebra to the Witt algebra.     

A note on number of orderings of higher level

The main aim of the paper is to find an explicit formula for the number of orderings of higher level. Among others, we discuss the relationship between the number of orderings of higher level and the number of orderings of level 1. We also construct a field with a given possible number of orderings of higher level. Received: 10 December 2004     

Two codes related to hermitian forms

Hermitian forms, exponential sums and linear algebra give us the opportunity to construct two trace-codes and obtain their parameters.     

A computation of the Witt index for rational quadratic forms

Summary This paper adds the finishing touches to an algorithmic treatment of quadratic forms over the rational numbers. The Witt index of a rational quadratic form is explicitly computed. When combined with a recent adjustment in the Haase invariants, this gives a complete set of invariants for rational quadratic forms, a set which can be computed and which respects all of the standard natural operations (including the tensor product) for quadratic forms. The overall approach does not use (at least explicitly) anyp-adic methods, but it does give the Witt ring of thep-adics as well as the Witt ring of the rationals.     

Indecomposable higher Chow cycles on Jacobians

We construct some natural indecomposable elements of with trivial regulator, and in particular, prove that is uncountable for C a generic curve or a generic hyperelliptic curve of genus . Received: 10 October 2000 / in final form: 12 June 2001 / Published online: 1 February 2002     

Motivic equivalence of quadratic forms. II

Let F be a field of characteristic ≠2 and φ be a quadratic form over F. By X _φ we denote the projective variety given by the equation φ=0. For each positive even integer d≥8 (except for d=12) we construct a field F and a pair φ, ψ of anisotropic d-dimensional forms over F such that the Chow motives of X _φ and X _ψ coincide but . For a pair of anisotropic (2 ^{n -1})-dimensional quadrics X and Y, we prove that existence of a rational morphism Y→X is equivalent to existence of a rational morphism Y→X. Received: 27 September 1999 / Revised version: 27 December 1999     

Higher trace forms and essential dimension in central simple algebras

We show that the essential dimension of a finite-dimensional central simple algebra coincides with the essential dimension of its r-linear trace form, for any r ≥ 3. Received: 15 March 2006     

Weighted sum of the extensions of the representations of quadratic forms

Let L, N and M be positive definite integral \({\mathbb{Z}}\) -lattices. In this paper, we show some relation between the weighted sum of representations of L and N by gen(M) and the weighted sum of extensions of \(\tilde M_{\tilde \sigma}\) in the gen(M _σ) via N _η when M is even and gcd(dL, dM) =  1. As a consequence of the particular case when M is even unimodular, we recapture the Böcherer formula (13) in (Böcherer, Maths Z 183:21–46, 1983) for the relation of the Fourier coefficients between Eisenstein series and Jacobi–Eisenstein series.     

Annihilating polynomials of excellent quadratic forms

If φ is an excellent form, then it is possible to use the dimensions of the higher complements of φ to obtain an annihilating polynomial of φ of low degree. The main result of this paper is the construction of such a polynomial with the help of methods from the theory of generic splitting of quadratic forms. Received: 23 April 2007     

HIGHER DEGREE HYPERBOLIC FORMS

Abstract

We define higher degree hyperbolic forms, analogous to the quadratic hyperbolic forms. We prove the following descent result. Let f be a form of degree d ≥ 3 over a field F of characteristic 0, and let K|f be a field extension. Then if f is equivalent over K to a hyperbolic form, f must already be equivalent to it over F. We also prove that in the monoid of equivalence classes of forms defined over F of a fixed degree d ≥ 3, under the tensor product, the submonoid generated by the equivalence classes of the hyperbolic forms is free. The proofs of these results involve the calculation of the centres and the Lie algebras of the higher degree hyperbolic forms. For the convenience of the reader we expound some of Harrison's seminal paper [5].     

Note on the degree of concavity of the determinant on sets of positive definite quadratic forms

An upper bound is determined for the for whichD(f) is a concave function off, wheref ranges over Minkowski-reduced positive definite quadratic forms inn variables with diagonal coefficients unity andD(f) denotes the determinant off In answer to a question of C E Nelson, it is shown thatD(f) is not concave in the casen 4     

Homogeneous tri-additive forms and derivations

Gleason [A.M. Gleason, The definition of a quadratic form, Amer. Math. Monthly 73 (1966) 1049-1066] determined all functionals Q on K-vector spaces satisfying the parallelogram law Q(x+y)+Q(x-y)=2Q(x)+2Q(y) and the homogeneity Q(λx)=λ²Q(x). Associated with Q is a unique symmetric bi-additive form S such that Q(x)=S(x,x) and 4S(x,y)=Q(x+y)-Q(x-y). Homogeneity of Q corresponds to that of S: S(λx,λy)=λ²S(x,y). The associated S is not necessarily bi-linear.Let V be a vector space over a field K, char(K)≠2,3. A tri-additive form T on V is a map of V³ into K that is additive in each of its three variables. T is homogeneous of degree 3 if T(λx,λy,λz)=λ³T(x,y,z) for all .We determine the structure of tri-additive forms that are homogeneous of degree 3. One of the keys to this investigation is to find the general solution of the functional equation

F(t)+t³G(1/t)=0,     

On the generic splitting of quadratic forms in characteristic 2

In [9] and [10] Knebusch established the basic facts of generic splitting theory of quadratic forms over a field of characteristic different from 2. This paper is related to [11] and [13] where Knebusch and Rehmann generalized partially this theory to a field of characteristic 2. More precisely, we begin with a complete characterization of quadratic forms of height 1 (we don't exclude anisotropic quadratic forms with quasi-linear part of dimension at least 1). This allows us to extend the notion of degree to characteristic 2. We prove some results on excellent forms and splitting tower of a quadratic form. Some results on quadratic forms of height 2 and degree 1 or 2 are given. Received: 6 March 2000; in final form: 5 October 2001 / Published online: 17 June 2002     

Sesquilinear forms over rings with involution

Many classical results concerning quadratic forms have been extended to Hermitian forms over algebras with involution. However, not much is known in the case of sesquilinear forms without any symmetry property. The present paper will establish a Witt cancellation result, an analogue of Springer’s theorem, as well as some local–global and finiteness results in this context.