Arbitrary Quadratic Form Research Articles

Upper bounds for the rank of powers of quadrics

We establish an upper bound for the rank of every power of an arbitrary quadratic form. Specifically, for any s∈N, we prove that the s-th power of a quadratic form of rank n grows as ns. Furthermore, we demonstrate that its rank is subgeneric for all n>(2s−1)2.

Isotropy indices of Pfister multiples in characteristic 2

Let F be a field of characteristic 2, π an n-fold bilinear Pfister form over F and φ an arbitrary quadratic form over F. In this note, we investigate Witt index, defect, total isotropy index and higher isotropy indices of φ and π⊗φ and prove relations among the indices of these two forms over certain field extensions.

The birational composition of arbitrary quadratic form with binary quadratic form

Let f(X) and g(Y) be nondegenerate quadratic forms of dimensions m and n respectively over a field K, charK ≠ 2. Herein, the problem of the birational composition of f(X) and g(Y) is considered, namely, the condition is established when the product f(X)g(Y) is birationally equivalent over K to a quadratic form h(Z) over K of dimension m + n? The main result of this paper is the complete solution of the problem of the birational composition for quadratic forms f(X) and g(Y) over a field K when m = 2. The sufficient and necessary conditions for the existence of birational composition h(Z) for quadratic forms f(X) and g(Y) over a field K for m = 2 are obtained. The set of quadratic forms is described which can be considered as h(Z) in this case.

Nonlinear wave equation in a cosmological Kaluza Klein spacetime

We prove the global existence for the Cauchy problem of a semi-linear wave equation with an arbitrary quadratic form in a cosmological Kaluza–Klein spacetime, which is the product of a Milne universe with S1. The data prescribed on t = t0 are small, and t0 > 0 can arbitrarily approach the t = 0 singularity. Our proof relies on a decomposition of the wave equation into the zero and non-zero modes, and a crucial nonlinear structure after this decomposition is observed. In addition, we introduce various modified energies and the associated energy identities according to different expectations of decaying estimates.

Similarity of quadratic and symmetric bilinear forms in characteristic 2

We say that a field extension L/F has the descent property for isometry (resp. similarity) of quadratic or symmetric bilinear forms if any two forms defined over F that become isometric (resp. similar) over L are already isometric (resp. similar) over F. The famous Artin–Springer theorem states that anisotropic quadratic or symmetric bilinear forms over a field stay anisotropic over an odd degree field extension. As a consequence, odd degree extensions have the descent property for isometry of quadratic as well as symmetric bilinear forms. While this is well known for nonsingular quadratic forms, it is perhaps less well known for arbitrary quadratic or symmetric bilinear forms in characteristic 2. We provide a proof in this situation. More generally, we show that odd degree extensions also have the descent property for similarity. Moreover, for symmetric bilinear forms in characteristic 2, one even has the descent property for isometry and for similarity for arbitrary separable algebraic extensions. We also show Scharlau’s norm principle for arbitrary quadratic or bilinear forms in characteristic 2.

On prime values of binary quadratic forms with a thin variable

In this paper, we generalize the result of Fouvry and Iwaniec dealing with prime values of the quadratic form x 2 + y 2 with one input restricted to a thin subset of the integers. We prove the same result with an arbitrary primitive positive definite binary quadratic form. In particular, for any positive definite binary quadratic form F and binary linear form G, there exist infinitely many ℓ , m ∈ Z such that both F ( ℓ , m ) and G ( ℓ , m ) are primes as long as there are no local obstructions.

Equivariant Gauss sum of finite quadratic forms

Abstract The classical quadratic Gauss sum can be thought of as an exponential sum attached to a quadratic form on a cyclic group. We introduce an equivariant version of Gauss sum for arbitrary finite quadratic forms, which is an exponential sum twisted by the action of the orthogonal group. We prove that simple arithmetic formulas hold for some basic classes of quadratic forms. In applications, such invariant appears in the dimension formula for certain vector-valued modular forms.

Vahlen groups defined over commutative rings

Elements of a Vahlen group are $$2 \times 2$$ matrices with entries in a Clifford algebra satisfying some conditions. They play a central role in the theory of higher dimensional harmonic automorphic forms. Traditionally they have come in both ordinary and paravector type and have been defined (over Clifford algebras) over the real or complex numbers. We extend the definition of both types to be over a commutative ring with an arbitrary quadratic form. We show that they are indeed groups and identify in each case the group as the pin group, spin group, or another subgroup of the Clifford group. Under some mild conditions, for both types we show the equivalence of our definition with a suitably generalised version of the two standard definitions.

Characteristic function of a quadratic form formed by correlated complex Gaussian variables

An exact closed-form analytical expression is obtained for a one-dimensional characteristic function of an arbitrary quadratic form formed by statistically dependent Gaussian random variables with nonzero means. Necessary and sufficient conditions under which an arbitrary quadratic form has the generalized χ2 distribution are formulated. Parameters of the distribution of an arbitrary quadratic form are analyzed for particular cases.

The weak isotropy of quadratic forms over field extensions

The weak isotropy index (or equivalently, sublevel) of arbitrary quadratic forms is studied. Its relationship to the level of a form is investigated. The problem of determining the set of values of the weak isotropy index of a form as it ranges over field extensions is addressed, with both admissible and inadmissible numbers being determined. An analogous investigation with respect to the level of a form is also undertaken. A treatment of forms for which the above invariants coincide concludes this article, with some recently-raised questions being resolved.

On lattice sums and Wigner limits

Wigner limits are given formally as the difference between a lattice sum, associated to a positive definite quadratic form, and a corresponding multiple integral. To define these limits, which arose in work of Wigner on the energy of static electron lattices, in a mathematically rigorous way one commonly truncates the lattice sum and the corresponding integral and takes the limit along expanding hypercubes or other regular geometric shapes. We generalize the known mathematically rigorous two- and three-dimensional results regarding Wigner limits, as laid down in [3], to integer lattices of arbitrary dimension. In doing so, we also resolve a problem posed in [6, Chapter 7]. For the sake of clarity, we begin by considering the simpler case of cubic lattice sums first, before treating the case of arbitrary quadratic forms. We also consider limits taken along expanding hyperballs with respect to general norms, and connect with classical topics such as Gauss's circle problem. Appendix A is included to recall certain properties of Epstein zeta functions that are either used in the paper or serve to provide perspective.

OPERATOR CALCULUS AND INVERTIBLE CLIFFORD APPELL SYSTEMS: THEORY AND APPLICATION TO THE n-PARTICLE FERMION ALGEBRA

Motivated by evolution equations on Clifford algebras and illustrated with the n-particle fermion algebra, a theory of invertible left- and right-Appell systems is developed for Clifford algebras of an arbitrary quadratic form. This work extends and clarifies the authors' earlier work on Clifford Appell systems, operator calculus, and operator homology/cohomology. A direct connection is also shown between blade factorization algorithms and the construction of Appell systems in these algebras.

Quadratic Form made a Perfect Power by a New Composition Theorem on Arbitrary Quadratic Forms

This paper deals with the diophantine equation Q(x_1,\,x_2,\,\ldots,\,x_m)= y^n , where m and n are arbitrary positive integers and Q(x_1,\,x_2,\,\ldots,\,x_m) is an arbitrary quadratic form in the m variables x_1,\,x_2,\,\ldots,\,x_m . While solutions of special cases of this equation have been published earlier, the general equation of this type has not been solved till now. To solve this equation, we first show that, given an arbitrary quadratic form Q(x_1,\,x_2,\,\ldots,\,x_m) in m variables, there exists a composition formula Q(u_i)\,Q^2(v_i)=Q(w_i) where u_i and v_i ( i= 1,\,2,\,\ldots,\,m ) are arbitrary variables and the w_i ( i=1,\,2,\,\ldots,\,m ) are cubic forms in the variables u_i and v_i ( i=1,\,2,\,\ldots,\,m ). This is a new composition formula, different from the standard composition formulae of the type Q(u_i)Q(v_i)=Q(w_i) which are known for certain classes of quadratic forms. As the equation Q(x_i)=y^n is not always solvable, we prove a theorem giving a necessary and sufficient condition for its solvability. We use the aforementioned composition formula to obtain parametric solutions of the equation Q(x_i)=y^n , and also give some numerical examples.

ANNIHILATING IDEALS OF QUADRATIC FORMS OVER LOCAL AND GLOBAL FIELDS

We study annihilating polynomials and annihilating ideals for elements of Witt rings for groups of exponent 2. With the help of these results and certain calculations involving the Clifford invariant, we are able to give full sets of generators for the annihilating ideal of both the isometry class and the equivalence class of an arbitrary quadratic form over a local field. By applying the Hasse–Minkowski theorem, we can then achieve the same for an arbitrary quadratic form over a global field.

Symmetries of quadratic form classes and of quadratic surd continued fractions. Part II: Classification of the periods’ palindromes

According to a theorem by Lagrange, the continued fractions of quadratic surds are periodic. Their periods may have different types of symmetries. This work relates these types of symmetries to the symmetries of the classes of the corresponding indefinite quadratic forms. This allows classifying the periods of quadratic surds and simultaneously finding the symmetry type of the class of an arbitrary indefinite quadratic form and the number of its integer points contained in each domain of the Poincare tiling of the de Sitter world, introduced in Part I of this paper. Moreover, we obtain the same result for every class of forms representing zero, i.e., when the quadratic surds are replaced by rational, using the finite continued fraction obtained from a special representative of that class. Finally, we show the relation between the reduction procedure for indefinite quadratic forms defined by continued fractions and the classical reduction theory, which acquires a geometric description by the results in Part I.

Representation of integers by isotropic ternary quadratic forms

The discrete ergodic method is used to find a complete solution of the problem of the asymptotic behaviour of the number of representations of integers by an arbitrary integral isotropic ternary quadratic form both over regions on the corresponding surface and over the residue classes of a given modulus.

The Balian–Low theorem for the symplectic form on R2d

In this paper we extend the Balian–Low theorem, which is a version of the uncertainty principle for Gabor (Weyl–Heisenberg) systems, to functions of several variables. In particular, we first prove the Balian–Low theorem for arbitrary quadratic forms. Then we generalize further and prove the Balian–Low theorem for differential operators associated with a symplectic basis for the symplectic form on R2d.

Asgeirsson's Mean Value Theorem and Related Identities

Asgeirsson's mean value theorem states that if u satisfies the ultrahyperbolic differential equation (Δx−Δy)u=0 in a neighborhood of the convex compact set KR={(x, y); x, y∈Rν, |x|+|y|⩽R}, then 〈u, f〉=0 where f is the difference between the surface measures on the spheres {(x, 0); x∈Rν, |x|=R} and {(0, y); y∈Rν, |y|=R}. We extend this to solutions of the inhomogeneous equation by proving that f=(Δx−Δy)μR where μR=12χ(1−ν)/2+(πAR/4R2), with AR(x, y)=(|x|2−|y|2)2−2R2 (|x|2+|y|2)+R4. The distributions χa+ on R are defined by χa+(t)=ta+/Γ(a+1) when Rea>−1 and then continued analytically to all a∈C. This formula is closely related to the fundamental solution of the wave equation in Rν+1. Similar identities are given for arbitrary indefinite nonsingular real quadratic forms in R2ν. This work was originally motivated by the progressive solutions of the wave equation given by G. Friedlander and M. Riesz.

Local zeta functions of general quadratic polynomials

This paper is concerned with the kind of local zeta functions now often called Igusa’s local zeta functions: A simple closed form of such a zeta function for an arbitrary quadratic form, its variants, and an application are given.

Fundamentalsätze der metrischen Geometrie

For arbitrary quadratic forms, including the cases of characteristic 2 and of infinite dimensions, several affine-metric and projective-metric structures are considered, and the corresponding isomorphisms are determined. As an application, a general fundamental theorem of the miquelian circle geometry is proved.