Witt index for Galois Ring valued quadratic forms

Quadratic Forms over Fields Ii: Structure of the Witt Ring

On the First Witt Index of Quadratic Forms

We prove Hoffmann’s conjecture determining the possible values of the first Witt index of anisotropic quadratic forms of any given dimension. The proof makes use of the Steenrod type operations on the modulo 2 Chow groups constructed by P. Brosnan. Let F be a field of characteristic 6= 2. For an anisotropic quadratic form φ over F with dim(φ) ≥ 2, the first Witt index i1(φ) is the Witt index (i...

Computational problems for vector-valued quadratic forms

For R-vector spaces U and V we consider a symmetric bilinear map B : U×U → V . This then defines a quadratic map QB : U → V by QB(u) = B(u, u). Corresponding to each λ ∈ V ∗ is a R-valued quadratic form λQB on U defined by λQB(u) = λ · QB(u). B is definite if there exists λ ∈ V ∗ so that λQB is positive-definite. B is indefinite if for each λ ∈ V ∗, λQB is neither positive nor negative-semidefi...

-Valued Quadratic Forms and Quaternary Sequence Families

In this paper, -valued quadratic forms defined on a vector space over are studied. A classification of such forms is established, distinguishing -valued quadratic forms only by their rank and whether the associated bilinear form is alternating. This result is used to compute the distribution of certain exponential sums, which occur frequently in the analysis of quaternary codes and quaternary s...

Integer-Valued Quadratic Forms and Quadratic Diophantine Equations

We investigate several topics on a quadratic form Φ over an algebraic number field including the following three: (A) an equation ξΦ · ξ = Ψ for another form Ψ of a smaller size; (B) classification of Φ over the ring of algebraic integers; (C) ternary forms. In (A) we show that the “class” of such a ξ determines a “class” in the orthogonal group of a form Θ such that Φ ≈ Ψ ⊕Θ. Such was done in ...