Related papers: Quadratic Involutions on Binary Forms
In this work, we offer a historical stroll through the vast topic of binary quadratic forms. We begin with a quick review of their history and then an overview of contemporary algebraic developments on the subject.
In this paper, we study additively indecomposable quadratic forms over real biquadratic and simplest cubic fields. In particular, we show that over these fields, we can always find such a classical form in 2 variables, which differs from…
Counting integral binary quadratic forms with certain restrictions is a classical problem. In this paper, we count binary quadratic forms of fixed discriminant given restrictions on the size of their coefficients. We accomplish this by…
Let R be a ring. A construction method for flexible quadratic algebras with scalar involution over R is presented which unifies various classical constructions in the literature, in particular those to construct composition algebras.
We associate to every central simple algebra with involution of orthogonal type in characteristic two a totally singular quadratic form which reflects certain anisotropy properties of the involution. It is shown that this quadratic form can…
In this article we study the involutions of $\mathrm{O}(V,\mathrm{q})$, an orthogonal group for a vector space $V$ with quadratic form $\mathrm{q}$ over a field of characteristic 2. The classification proceeds by discussing conjugacy…
We study some properties of quadratic forms with values in a field whose underlying vector spaces are endowed with the structure of right vector spaces over a division ring extension of that field. Some generalized notions of isotropy,…
For quadratic spaces which represent 1 there is a characterization of hermitian compositions in the language of algebras-with-involutions using the even Clifford algebra. We extend this notion to define a generalized composition based on…
Quadratic descent of hermitian and skew hermitian forms over division algebras with involution of the first kind in arbitrary characteristic is investigated and a criterion, in terms of systems of quadratic forms, is obtained. A refined…
We classify conjugacy classes of involutions in the isometry groups of nondegenerate, symmetric bilinear forms over the field of two elements. The new component of this work focuses on the case of an orthogonal form on an even dimensional…
This dissertation presents a multifaceted look into the structural decomposition of permutation classes. The theory of permutation patterns is a rich and varied field, and is a prime example of how an accessible and intuitive definition…
The u-invariant of a field is the supremum of the dimensions of anisotropic quadratic forms over the field. We define corresponding u-invariants for hermitian and generalised quadratic forms over a division algebra with involution in…
It will be shown that transformations of order one on the Wiener space give rise to quadratic forms as exponents of change of variables formulas, and conversely every exponentially integrable quadratic form has a transformation of order one…
We investigate the quadratic descent of totally decomposable algebras with involution of orthogonal type in characteristic two. Both separable and inseparable extensions are included.
An involution is usually defined as a mapping that is its own inverse. In this paper, we study quaternion involutions that have the additional properties of distribution over addition and multiplication. We review formal axioms for such…
We discuss an unusual phenomenon in (integral) positive ternary quadratic forms. We also describe an interesting pairing of genera of ternary forms.
The structural constants of an evolution algebra is given by a quadratic matrix $A$. In this work we establish equivalence between nil, right nilpotent evolution algebras and evolution algebras, which are defined by upper triangular matrix…
We observe structure in the sequences of quotients and remainders of the Euclidean algorithm with two families of inputs. Analyzing the remainders, we obtain new algorithms for computing modular inverses and representating prime numbers by…
In this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class…
Both a general and a diagonal u-invariant for forms of higher degree are defined, generalizing the u-invariant of quadratic forms. Both old and new results on these invariants are collected.