Witt index for Galois Ring valued quadratic forms

We calculate the Witt index for Galois Ring valued quadratic forms. We obtain such index depending on the invariant that classifies nonsingular Galois Ring valued quadratic forms together with the type of the corresponding bilinear form (alternating or not).     

An invariant for quadratic forms valued in Galois Rings of characteristic 4

We introduce an invariant for nonsingular quadratic forms that take values in a Galois Ring of characteristic 4. This notion extends the invariant in ${\mathbb{Z}}_8$ for ${\mathbb{Z}}_4$-valued quadratic forms defined by Brown [E.H. Brown, Generalizations of the Kervaire invariant, Ann. of Math. (2) 95 (2) (1972) 368–383] and studied by Wood [J.A. Wood, Witt's extension theorem for mod four valued quadratic forms, Trans. Amer. Math. Soc. 336 (1) (1993) 445–461]. It is defined in the associated Galois Ring of characteristic 8. Nonsingular quadratic forms are characterized by their invariant and the type of the associated bilinear form (alternating or not).     

Integral bilinear forms, Coxeter transformations and Coxeter polynomials of finite posets

Linear algebra technique in the study of linear representations of finite posets is developed in the paper. A concept of a quadratic wandering on a class of posets I is introduced and finite posets I are studied by means of the four integral bilinear forms (1.1), the associated Coxeter transformations, and the Coxeter polynomials (in connection with bilinear forms of Dynkin diagrams, extended Dynkin diagrams and irreducible root systems are also studied). Bilinear equivalences between some of the forms are established and equivalences with the bilinear forms of Dynkin diagrams and extended Dynkin diagrams are discussed. A homological interpretation of the bilinear forms (1.1) is given and Z-bilinear equivalences between them are discussed. By applying well-known results of Bongartz, Loupias, and Zavadskij-Shkabara, we give several characterisations of posets I, with the Euler form weakly positive (resp. with the reduced Euler form weakly positive), and posets I, with the Tits form weakly positive.     

Representations by bilinear forms (and.p a)

Summary An expression is derived for the number of representations of one bilinear form by another (mod p^a). From this, an explicit formula for the number of such representations is obtained in the case where both forms have square nonsingular matrices (mod p). A related bilinear analog of a lemma of Siegel on representations by quadratic forms (mod p^a) is also proved. In memory of guido Castelnuovo, in the recurrence of the first centenary of his birth. Research supported by National Science Foundation Grants GP-2542 and GP-4015.     

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 four-dimensional Kerdock set over GF(3)

If m is an even integer and K = GF(q) is a field of characteristic 2, then there exists a set of q^m?1 alternating bilinear forms of degree m over K such that the difference of any two of the forms is nonsingular. Do such sets exist over fields of odd characteristic? This note constructs such a set in the smallest nontrivial case, namely, m = 4, q = 3.     

Berezin kernels and analysis on Makarevich spaces

Following ideas of van Dijk and Hille we study the link which exists between maximal degenerate representations and Berezin kernels.We consider the conformal group Conf(V) of a simple real Jordan algebra V. The maximal degenerate representations π_s (s ε ℂ) we shall study are induced by a character of a maximal parabolic subgroup P¯ of Conf(V). These representations π_s can be realized on a space I_s of smooth functions on V. There is an invariant bilinear form ℬ_s on the space Is. The problem we consider is to diagonalize this bilinear form ℬ_s, with respect to the action of a symmetric subgroup G of the conformal group Conf(V). This bilinear form can be written as an integral involving the Berezin kernel B_v an invariant kernel on the Riemannian symmetric space G/K, which is a Makarevich symmetric space in the sense of Bertram. Then we can use results by van Dijk and Pevzner who computed the spherical Fourier transform of B_v. From these, one deduces that the Berezin kernel satisfies a remarkable Bernstein identity: D(_ν)B_ν=b(ν)B_ν+1, where D(_ν) is an invariant differential operator on G/K and b(ν) is a polynomial. By using this identity we compute a Hua type integral which gives the normalizing factor for an intertwining operator from I_−s to I_s. Furthermore, we obtain the diagonalization of the invariant bilinear form with respect to the action of the maximal compact group U of the conformal group Conf(V).     

Some elementary proofs of lower bounds in complexity theory

We introduce a new and easily applicable criterion called rank immunity for estimating the minimal number of multiplications needed to compute a set of bilinear forms in commuting variables. The result is obtained by an elimination argument after canonically embedding computations in a quotient ring R/I, where I is an appropriately chosen ideal that is left invariant under the eliminations. The criterion combines the well-known arguments based on elimination and on row rank, but in contrast to (for instance) column- and mixed-rank arguments it normally leads to better elementary estimates than were derivable in a uniform manner before.     

Analysis of a problem of Raikov with applications to barreled and bornological spaces

Raikov’s conjecture states that semi-abelian categories are quasi-abelian. A first counterexample is contained in a paper of Bonet and Dierolf who considered the category of bornological locally convex spaces. We prove that every semi-abelian category I admits a left essential embedding into a quasi-abelian category K_l(I) such that I can be recovered from K_l(I) by localization. Conversely, it is shown that left essential full subcategories I of a quasi-abelian category are semi-abelian, and a criterion for I to be quasi-abelian is given. Applied to categories of locally convex spaces, the criterion shows that barreled or bornological spaces are natural counterexamples to Raikov’s conjecture. Using a dual argument, the criterion leads to a simplification of Bonet and Dierolf’s example.     

Certain systems of bilinear forms whose minimal algorithms are all quadratic

In this paper we prove that every minimal algorithm for computing a bilinear form is quadratic. It is then easy to show that for certain systems of bilinear forms all minimal algorithms are quadratic. One such system is for computing the product of two arbitrary elements in a finite algebraic extension field. This result, together with the results of the author in (E. Feig, to appear) and those of Winograd (Theoret. Comput. Sci.8 (1979), 359–377), completely characterize all minimal algorithms for computing products in these fields.     

The relation of quadratic k-theory to hermitian k:-Theory

It is known that the Hermitian algebraic K-theory of a ring with antistructure is the fixed point theory of a certain involution on the algebraic K-theory of the ground ring. It is shown here that the (unitary) algebraic K-theory of nonsingular split quadratic forms over a ring with antistructure (R,α,u) is the same as the algebraic K-theory of nonsingular Hermitian forms over a related ring with antistructure (S?v). Specifically, S = R = R[d]/(d²) is the ring of dual numbers of R, the conjugation β is the antiautomorphism of S given by β{r + r’d) = α(r) - α(r’)d, and the unit of symmetry v= u(l + d). This makes it possible to regard split quadratic algebraic K-theory of R as a fixed point theory of an involution on algebraic K-theory of R. On the level of K_o (Grothendieck) groups, quadratic and split quadratic algebraic K-theory are the same. This is applied to show that the Kervaire-Arf invariant of skew-quadratic forms over Z (here (α,u) = (id,-l)) becomes identified with a generalized Hasse-Witt invariant over (Z, β,-(1+d)).     

Tensor products and discriminants of unital quadratic forms over commutative rings

A tensor product for unital quadratic forms is introduced which extends the product of separable quadratic algebras and is naturally associative and commutative. It admits a multiplicative functor vdis, the vector discriminant, with values in symmetric bilinear forms. We also compute the usual (signed) discriminant of the tensor product in terms of the discriminants of the factors. The orthogonal group scheme of a nonsingular unital quadratic formQ of even rank is isomorphic toZ ₂×SO(Q ⁰) whereQ ⁰ is the restriction of –Q to the space of trace zero elements. We use cohomology to interpret the action of separable quadratic algebras on unital quadratic forms, and to determine which forms of odd rank can be realized asQ ⁰.     

A necessary and sufficient condition for a class of nonsingular Galois NFSRs

Let n be a positive integer. An n-stage Galois NFSR is nonsingular if and only if the output sequences of each bit register are purely periodic. It is well known that a Galois NFSR used to build a stream cipher should be nonsingular. Recently, a useful concept that is the simplified feedback function is proposed for a Galois NFSR, which is a vectorial Boolean function. Generally, for a Galois NFSR, its simplified feedback function has less nonlinear terms than its traditional feedback function, and so is easier to analyze. Moreover, it has been shown that the nonsingularity of a Galois NFSR is decided by the invertibility of its simplified feedback function. Based on this observation, in this paper, we present a necessary and sufficient condition for the nonsingularity of a class of Galois NFSRs such that every component of its simplified feedback is linear or has a common nonlinear term.     

Eine Kongruenz für Gitter und ihre Anwendung auf 2k - Mannigfaltigkeiten in 4k - Mannigfaltigkeiten

In this article we consider nonsingular unimodular lattices. With every element, which is divisible by 2, we associate a so-called Brown invariant and prove a congruence relating the self-intersection number of the element, the signatur of the corresponding quadratic form and the Brown invariant, defined by this element. This result will be applied to study involutions on 4-dimensional manifolds.     

Equivalent conditions for noncentral generalized Laplacianness and independence of matrix quadratic forms

Let Y be an n×p multivariate normal random matrix with general covariance Σ_Y and W be a symmetric matrix. In the present article, the property that a matrix quadratic form Y^′WY is distributed as a difference of two independent (noncentral) Wishart random matrices is called the (noncentral) generalized Laplacianness (GL). Then a set of algebraic results are obtained which will give the necessary and sufficient conditions for the (noncentral) GL of a matrix quadratic form. Further, two extensions of Cochran’s theorem concerning the (noncentral) GL and independence of a family of matrix quadratic forms are developed.     

A note on biquadratic forms

It is shown that semidefinite quadratic forms in two by n variables are sums of squares of bilinear forms.     

Space of invariant bilinear forms

Let \({\mathbb {F}}\) be a field, V a vector space of dimension n over \({\mathbb {F}}\). Then the set of bilinear forms on V forms a vector space of dimension \(n^2\) over \({\mathbb {F}}\). For char \({\mathbb {F}}\ne 2\), if T is an invertible linear map from V onto V then the set of T-invariant bilinear forms, forms a subspace of this space of forms. In this paper, we compute the dimension of T-invariant bilinear forms over \({\mathbb {F}}\). Also we investigate similar type of questions for the infinitesimally T-invariant bilinear forms (T-skew symmetric forms). Moreover, we discuss the existence of nondegenerate invariant (resp. infinitesimally invariant) bilinear forms.     

On the number of anisotropic simple submodules in modules with a form

Let G be a finite group, and V a finite-dimensional semisimple G-module over a finite field. Assume that V is endowed with a nonsingular bilinear form which is symmetric or symplectic, and which is invariant under the action of G. In this setting, we compute the number of anisotropic simple submodules of V.Received: 25 May 2004