Splitting Patterns and Invariants of Quadratic Forms

Let φ be an anisotropic quadratic form over a field F of characteristic not 2. The splitting pattern of φ is defined to be the increasing sequence of nonnegative integers obtained by considering the Witt indices i_W(φ_k) of φ over K where K ranges over all field extensions of F. Restating earlier results by HURRELBRINK and REHMANN , we show how the index of the Clifford algebra of φ influences the splitting pattern. In the case where F is formally real, we investigate how the signatures of φ influence the splitting behaviour. This enables us to construct certain splitting patterns which have been known to exist, but now over much “simpler” fields like formally real global fields or ?(t). We also give a full classification of splitting patterns of forms of dimension less than or equal to 9 in terms of properties of the determinant and Clifford invariant. Partial results for splitting patterns in dimensions 10 and 11 are also provided. Finally, we consider two anisotropic forms φ and φ of the same dimension m with φ ? ? φ ∈ Iⁿ F and give some bounds on m depending on n which assure that they have the same splitting pattern.     

Anisotropic Forms of Higher Degree Under Finite Dimensional Field Extensions

Let F be a field of characteristic 0 or greater than d. We investigate anisotropic forms of degree d > 2 and their behaviour under field extensions of degree coprime to d. A form of degree d over F which permits composition or Jordan composition remains anisotropic under any field extension of degree coprime to d. Other classes of forms are constructed which stay anisotropic under such extensions.     

Generic Extensions of Prinjective Modules

Let be an algebraically closed field and let be a finite partially ordered set of finite prinjective type. We study generic extensions of prinjective modules over the incidence -algebra of . We prove that there exist generic extensions of prinjective -modules and describe properties of the monoid of generic extensions.Partially supported by Polish KBN Grant 5 P0 3A 015 21.     

Quadratic Extensions of Flag-transitive Planes

In this article, the finite translation planes which are quadratic extensions of flag-transitive subplanes are classified as either Desarguesian or Hall.     

Distinguished Representations for Quadratic Extensions

Let K be a quadratic extension of a field k which is either local field or a finite field. Let G be an algebraic group over k. The aim of the present paper is to understand when a representation of G(K) has a G(k) invariant linear form. We are able to accomplish this in the case when G is the group of invertible elements of a division algebra over k of odd index if k is a local field, and for general connected groups over finite fields.     

Hyperbolic Involutions and Quadratic Extensions

This is a variation on a theme of Bayer-Fluckiger, Shapiro, and Tignol related to hyperbolic involutions. More precisely, criteria for the hyperbolicity of involutions of quadratic extensions of simple algebras and involutions of the form σ ? τ and σ ? ρ, where σ is an involution of a central simple algebra A, τ is the nontrivial automorphism of a quadratic extension of the center of A, and ρ is an involution of a quaternion algebra are obtained.     

Generic Fiber of Power Series Ring Extensions

Let D be a Noetherian domain containing a field, a ∈ D a nonzero nonunit, and z an indeterminate over D. We prove that the generic fiber of the extension D[[z]]? D[1/a][[z]] has dimension ≥ dim(D/aD).     

Hilbert Functions and Generic Forms

Let A be a homogeneous K-algebra where K is a field of characteristic 0, and h A a generic form. We bound the Hilbert function H(A/(h),-) in terms of H(A,-) which extends the bound given by M.Green for generic linear forms. We apply this to some conjectures from Higher Castelnuovo Theory and Cayley-Bacharach Theory.     

The Groups LS and Morphisms of Quadratic Extensions

For a morphism of quadratic extensions of antistructures, groups similar to the groups of obstructions to splitting along one-sided submanifolds are defined. These groups are a natural generalization of the splitting obstruction groups. The results obtained open additional possibilities for constructing groups and natural maps in L-theory.     

Association Schemes of Quadratic Forms and Symmetric Bilinear Forms

Let X _n and Y _n be the sets of quadratic forms and symmetric bilinear forms on an n-dimensional vector space V over , respectively. The orbits of GL _n( ) on X _n × X _n define an association scheme Qua(n, q). The orbits of GL _n( ) on Y _n × Y _n also define an association scheme Sym(n, q). Our main results are: Qua(n, q) and Sym(n, q) are formally dual. When q is odd, Qua(n, q) and Sym(n, q) are isomorphic; Qua(n, q) and Sym(n, q) are primitive and self-dual. Next we assume that q is even. Qua(n, q) is imprimitive; when (n, q) (2,2), all subschemes of Qua(n, q) are trivial, i.e., of class one, and the quotient scheme is isomorphic to Alt(n, q), the association scheme of alternating forms on V. The dual statements hold for Sym(n, q).     

On Quadratic Forms Between Spheres

For a quadratic form f: S^m Sⁿ, we first estimate the dimensions of its preimages and determine the cases when the upper bound is achieved. Combining results of Wood and Yiu, we then derive a complete classification of f when m 2n-2. By using orthogonal multiplication, we also introduce a new construction of quadratic forms between spheres, generalizing the classical Hopf construction.     

Automorphisms of Association Schemes of Quadratic Forms over a Finite Field of Characteristic Two

Let X_n denote the set of quadratic forms in n variables over a finite field F_q of characteristic 2, and X_n the association scheme on X_n by defining the relations with respect to the type of quadratic forms. We prove that every automorphism of X_n is of the form X P^tXP + Y for all X X_n, where P GL_n(F_q), is an automorphism of F_q, and Y X_n.2000 Mathematics Subject Classification: 05E30Supported by the National Natural Science Foundation of China (10271039), Hobei Natural Science Foundation (102132) and Hebei Education Committee.     

特征2的有限域上二次型间的表示个数

本文利用特征2有限域Fq上奇异正交几何理论给出Fq上二次型的型与型间具有秩k的表示的个数,并由此得到一般计数公式及其q超几何级数表达式.     

Field Extensions and Extensions of Linear Differential Operators

Let K R P be a tower of fields, N be a P-module, and : R N be a K-linear differential operator. The aim of this paper is to investigate whether the operator has an extension to P, i.e. if these exists a differential operator : P N such that |_R = . The results of this paper were published in Russian in Mat. Zametki 30(2) (1981), 237–248.