Quadratic descent of hermitian forms

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 result is also obtained for hermitian (resp. skew hermitian) forms over a quaternion algebra with symplectic (resp. orthogonal) involution.     

On the 3-Pfister Number of Quadratic Forms

For a field F of characteristic different from 2, containing a square root of -1, endowed with an F^×2-compatible valuation v such that the residue field has at most two square classes, we use a combinatorial analogue of the Witt ring of F to prove that an anisotropic quadratic form over F with even dimension d, trivial discriminant, and Hasse–Witt invariant can be written in the Witt ring as the sum of at most (d²)/8 3-fold Pfister forms.     

Prime Producing Quadratic Polynomials and Class Number One or Two

Let d≡ 5 mod 8 be a positive square-free integer and let h(d) be the class number of the real quadratic field ℚ(√d). Let p be a divisor of d = pq and let . Assume that is prime or equal to 1 for all integers x with 0≤x<W. Under the assumption that the Riemann hypothesis is true, we prove that if , then h(d) < 2. Furthermore we show that h(d)< 2 implies d < 4245. In the case when there exists at least one split prime less than W, we prove the following results without any assumptions on the Riemann hypothesis. If then h< 2 or h = 4. If , then h≤ 2, h = 4 or h = 2^t−2, where t is the number of primes dividing d. In the case when h = 2^t−2 we have , where φ = 2 or 4. 2000 Mathematics Subject Classification: Primary–11R29     

Quadratic forms connected with Fourier coefficients of Maass cusp forms

For the normalized Fourier coefficients of Maass cusp forms λ(n) and the normalized Fourier coefficients of holomorphic cusp forms a(n), we give the bound of ∑m12+m22+m32≤xλ(m12+m22+m32)Λ(m12+m22+m32) and ∑m12+m22+m32≤xa(m12+m22+m32)Λ(m12+m22+m32).     

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.     

Representations of integers by certain positive definite binary quadratic forms

We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n=x ²+Ny ² for a squarefree integer N.      

On the definity of quadratic forms subject to linear constraints

A short proof is given of the necessary and sufficient conditions for the positivity and nonnegativity of a quadratic form subject to linear constraints.     

Quadratic residues and non-residues for infinitely many Piatetski-Shapiro primes

In this paper, we prove a quantitative version of the statement that every nonempty finite subset of N+ is a set of quadratic residues for infinitely many primes of the form [nc] with 1 ≤ c ≤243/205. Correspondingly, we can obtain a similar result for the case of quadratic non-residues under reasonable assumptions. These results generalize the previous ones obtained by Wright in certain aspects.     

On the Representation of Numbers by the Direct Sums of Some Binary Quadratic Forms

The systems of bases are constructed for the spaces of cusp forms and . Formulas are obtained for the number of representations of a positive integer by the sum of k binary quadratic forms of the kind , of the kind and of the kind .     

Small Solutions of Quadratic Diophantine Equations

We consider quadratic diophantine equations of the shape for a polynomial Q(X₁, ..., X_s) Z[X₁, ..., X_s] of degree 2.Let H be an upper bound for the absolute values of the coefficientsof Q, and assume that the homogeneous quadratic part of Q isnon-singular. We prove, for all s 3, the existence of a polynomialbound _s(H) with the following property: if equation (1) hasa solution x Z^s at all, then it has one satisfying For s = 3 and s = 4 no polynomial bounds _s(H) were previouslyknown, and for s 5 we have been able to improve existing boundsquite significantly. 2000 Mathematics Subject Classification11D09, 11E20, 11H06, 11P55.     

用正交变换化实二次型为标准形的同步求解问题

作为《关于矩阵的特征值与特征向量同步求解问题》的续篇,利用其给出的方法,证明了新的定理.通过对实对称矩阵进行行列互逆变换,同步求出二次型的标准形及正交变换阵,简化了复杂的施密特正交化法,较好地解决了二次型标准形与正交变换阵同步求解问题.     

Elliptic curves and positive definite ternary forms

For two given ternary quadratic formsf(x, y, z) andg( x, y, z), letr(f, n) andr(g, n) be the numbers of representations of n represented byf( x, y, z) and g( x, y, z) respectively. In this paper we study the following problem: when will we haver(f, n) =r(g, n) orr( f, n)≠r(g, n). Our method is to use elliptic curves and the corresponding new forms.     

Quadratic forms and Pfister neighbors in characteristic 2

We study Pfister neighbors and their characterization over fields of characteristic , where we include the case of singular forms. We give a somewhat simplified proof of a theorem of Fitzgerald which provides a criterion for when a nonsingular quadratic form is similar to a Pfister form in terms of the hyperbolicity of this form over the function field of a form which is dominated by . From this, we derive an analogue in characteristic of a result by Knebusch saying that, in characteristic , a form is a Pfister neighbor if its anisotropic part over its own function field is defined over the base field. Our result includes certain cases of singular forms, but we also give examples which show that Knebusch's result generally fails in characteristic for singular forms. As an application, we characterize certain forms of height in the sense of Knebusch whose quasi-linear parts are of small dimension. We also develop some of the basics of a theory of totally singular quadratic forms. This is used to give a new interpretation of the notion of the height of a standard splitting tower as introduced by the second author in an earlier paper.

     

Universal octonary diagonal forms over some real quadratic fields

In this paper, we will prove there are infinitely many integers n such that n ²— 1 is square-free and admits universal octonary diagonal quadratic forms. Received: November 2, 1998.     

A note on biquadratic forms

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

On bilinear forms in normal variables

Bilinear forms in normal variables when the matrices of the forms are rectangular are considered. Explicit expressions for the cumulants, joint cumulants and joint cumulants of bilinear and quadratic forms are given. Necessary and sufficient conditions are established for the independence of two bilinear forms as well as a bilinear and a quadratic form. Special cases are shown to agree with known results.     

The Problem on Class Numbers of Quadratic Number Fields

It is a survey of the problem on class numbers of quadratic number fields.