Positive definite <Emphasis Type="Italic">n</Emphasis>-regular quadratic forms

A positive definite integral quadratic form f is called n-regular if f represents every quadratic form of rank n that is represented by the genus of f. In this paper, we show that for any integer n greater than or equal to 27, every n-regular (even) form f is (even) n-universal, that is, f represents all (even, respectively) positive definite integral quadratic forms of rank n. As an application, we show that the minimal rank of n-regular forms has an exponential lower bound for n as it increases.     

Positive definite quadratic forms representing integers of the form <Emphasis Type="Italic">an</Emphasis><Superscript>2</Superscript>+<Emphasis Type="Italic">b</Emphasis>

For any subset S of positive integers, a positive definite integral quadratic form is said to be S-universal if it represents every integer in the set S. In this article, we classify all binary S-universal positive definite integral quadratic forms in the case when S=S _a ={an ²∣n≥2} or S=S _a,b ={an ²+b∣n∈ℤ}, where a is a positive integer and ab is a square-free positive integer in the latter case. We also prove that there are only finitely many S _a -universal ternary quadratic forms not representing a. Finally, we show that there are exactly 15 ternary diagonal S ₁-universal quadratic forms not representing 1.     

A calculus for branched spines of 3-manifolds

Let g[n] be the minimum number of squares whose sum represents all positive definite integral quadratic forms of rank n which are represented by sums of squares. In this article, we first discuss representations of integers by unimodular lattices. We then estimate the order of magnitude of the diameter of the 2-graph of unimodular lattices of rank n. Combining these results we prove g[n]=O(3^n/2n log n). We also provide a lower bound for g[n]. Finally, we discuss s-integrable lattices as an application of our method.Mathematics Subject Classification (2000): 11E12, 11H06The first author was partially supported by KRF research fund (2004-070-C00001).     

Integral quadratic forms and Dirichlet series

A Dirichlet series with multiplicative coefficients has an Euler product representation. In this paper we consider the special case where these coefficients are derived from the numbers of representations of an integer by an integral quadratic form. At first we suppose this quadratic form to be positive definite. In general the representation numbers are not multiplicative. Instead we consider the average number of representations over all classes in the genus of the quadratic form. And we consider only representations of integers of the form tk ² with t square-free. If we divide the average representation number for these integers by a suitable factor, we do get a multiplicative function. Using results from Siegel (Ann. Math. 36:527–606, 1935), we derive a uniform expression for the Euler product expansion of the corresponding Dirichlet series. As a special case, we consider the standard quadratic form in n variables corresponding to the identity matrix. Here we use results from Shimura (Am. J. Math. 124:1059–1081, 2002). For 2≤n≤8, the genus of this particular quadratic form contains only one class, and this leads to a rather simple expression for the Dirichlet series, where the coefficients are just the number of representations of a square as the sum of n squares. Finally we consider the indefinite case, where we can get results similar to the definite case.     

On indecomposable definite hermitian forms

In this paper, for any given natural numbersn anda, we can construct explicitly positive definite indecomposable integral Hermitian forms of rankn over with discriminanta, with the following ten exceptions:n=2,a=1, 2, 4, 10;n=3,a=1, 2, 5;n=4,a=1;n=5,a=1; andn=7,a=1. In the exceptional cases there are no forms with the desired properties. The method given here can be applied to solving the problem of constructing indecomposable positive definite HermitianR _m -lattices of any given rankn and discriminanta, whereR _m is the ring of algebraic integers in an imaginary quadratic field with class number unity.     

An additive problem in the Fourier coefficients of cusp forms

 We establish an estimate on sums of shifted products of Fourier coefficients coming from holomorphic or Maass cusp forms of arbitrary level and nebentypus. These sums are analogous to the binary additive divisor sum which has been studied extensively. As an application we derive, extending work of Duke, Friedlander and Iwaniec, a subconvex estimate on the critical line for L-functions associated to character twists of these cusp forms. Received: 2 October 2001 / Revised version: 9 September 2002 / Published online: 28 March 2003 Mathematics Subject Classification (2000): Primary 11F30, 11F37; Secondary 11M41.     

On locally primitively universal quadratic forms

The Ramanujan Journal - A positive definite integral quadratic form is said to be almost (primitively) universal if it (primitively) represents all but at most finitely many positive integers. In...     

A notion of rank in set theory without choice

 Starting from the definition of `amorphous set' in set theory without the axiom of choice, we propose a notion of rank (which will only make sense for, at most, the class of Dedekind finite sets), which is intended to be an analogue in this situation of Morley rank in model theory. Received: 22 September 2000 / Revised version: 14 May 2002 Published online: 19 December 2002 The research of the first author was supported by the SERC. Mathematics Subject Classification (2000): 03E25 Key words or phrases: Rank – Degree – Amorphous     

A local–global principle for algebras with involution and hermitian forms

 Weakly hyperbolic involutions are introduced and a proof is given of the following local–global principle: a central simple algebra with involution of any kind is weakly hyperbolic if and only if its signature is zero for all orderings of the ground field. Also, the order of a weakly hyperbolic algebra with involution is a power of two, this being a direct consequence of a result of Scharlau. As a corollary an analogue of Pfister's local–global principle is obtained for the Witt group of hermitian forms over an algebra with involution. Received: 29 October 2001; in final form: 9 August 2002 / Published online: 16 May 2003 Mathematics Subject Classification (2000): 16K20, 11E39     

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.     

Cohomological invariants of quaternionic skew-hermitian forms

We define a complete system of invariants e _n,Q ,n ≥ 0 for quaternionic skew-hermitian forms, which are twisted versions of the invariants e _n for quadratic forms. We also show that quaternionic skew-hermitian forms defined over a field of 2-cohomological dimension at most 3 are classified by rank, discriminant, Clifford invariant and Rost invariant. Received: 30 April 2006     

The minimal height of quadratic forms of given dimension

Given an arbitrary n, we consider anisotropic quadratic forms of dimension n over all fields of characteristic ≠ 2 and prove that the height of an n-dimensional excellent form (depending on n only) is the (precise) lower bound of the heights of all forms of dimension n. The second and third authors were partially supported by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00287, KTAGS. The James D.Wolfensohn Fund and The Ellentuck Fund support is acknowledged by the second author. Received: 9 December 2005     

Quaternary universal forms over \mathbf{Q}(\sqrt{13})

Let be a real quadratic field over Q with m a square-free positive rational integer and be the integer ring in F. A totally positive definite integral n-ary quadratic form f=f(x ₁,…,x _n )=∑_1≤i,j≤n α _ij x _i x _j ( ) is called universal if f represents all totally positive integers in . Chan, Kim and Raghavan proved that ternary universal forms over F exist if and only if m=2,3,5 and determined all such forms. There exists no ternary universal form over real quadratic fields whose discriminants are greater than 12. In this paper we prove that there are only two quaternary universal forms (up to equivalence) over . For the proof of universality we apply the theory of quadratic lattices.      

Congruence of minimal surfaces and higher fundamental forms

 We show that minimal surfaces in space forms are determined, up to ambient isometries, by the induced metric and certain invariants which are defined in terms of the higher fundamental forms and the complex structure. Received: 6 May 2002 / Revised version: 5 July 2002 Current address: Institut für Mathematik, Universit?t Augsburg, Universit?tsstrasse 12, D-86135 Augsburg, Germany. e-mail: Theodoros.Vlachos@Math.Uni-Augsburg.De The author was supported by the Alexander von Humboldt Foundation Mathematics Subject Classification (2000): 53A10, 53C42 Acknowledgements. This work was written during the author's stay at the University of Augsburg as a research fellow of the Alexander von Humboldt Foundation. I wish to express my gratitude to the Alexander von Humboldt Foundation for its generous support. Moreover, I wish to thank Professor J.H. Eschenburg for many fruitful and stimulating conversations.     

Multiplicative factorizations of integral representations by quadratic forms

Multiplicative dependence of integral representations of integers by positive definite quadratic forms in an odd number of variables on square factors of the represented numbers is studied. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 15–55. Translated by A. S. Goloubeva.     

Convexity Conditions and the Legendre-Fenchel Transform for the Product of Finitely Many Positive Definite Quadratic Forms

While the product of finitely many convex functions has been investigated in the field of global optimization, some fundamental issues such as the convexity condition and the Legendre-Fenchel transform for the product function remain unresolved. Focusing on quadratic forms, this paper is aimed at addressing the question: When is the product of finitely many positive definite quadratic forms convex, and what is the Legendre-Fenchel transform for it? First, we show that the convexity of the product is determined intrinsically by the condition number of so-called ‘scaled matrices’ associated with quadratic forms involved. The main result claims that if the condition number of these scaled matrices are bounded above by an explicit constant (which depends only on the number of quadratic forms involved), then the product function is convex. Second, we prove that the Legendre-Fenchel transform for the product of positive definite quadratic forms can be expressed, and the computation of the transform amounts to finding the solution to a system of equations (or equally, finding a Brouwer’s fixed point of a mapping) with a special structure. Thus, a broader question than the open “Question 11” in Hiriart-Urruty (SIAM Rev. 49, 225–273, 2007) is addressed in this paper.     

Systems of quadratic Diophantine inequalities and the value distribution of quadratic forms

Let Q ₁,…,Q _r be quadratic forms with real coefficients. We prove that the set is dense in , provided that the system Q ₁(x) = 0,…,Q _r (x) = 0 has a nonsingular real solution and all forms in the real pencil generated by Q ₁,…,Q _r are irrational and have rank larger than 8r. Moreover, we give a quantitative version of the above assertion. As an application we study higher correlation functions of the value distribution of a positive definite irrational quadratic form. Author’s address: Institut für Statistik, Technische Universit?t Graz, A-8010 Graz, Austria     

Real quadratic fields with class number divisible by 3

 We shall show that the number of real quadratic fields whose absolute discriminant is ≤ x and whose class number is divisible by 3 is improving the existing best known bound of K. Chakraborty and R. Murty. Received: 7 January 2003 Published online: 19 May 2003 This work was supported by Korea Research Foundation Grant KRF-2002-003-C00001. Mathematics Subject Classification(2000): 11R11, 11R29     

On indecomposable definite unimodular hermitian forms

For any natural numbersm andn≥17 we can construct explicitly indecomposable definite unimodular normal Hermitian lattices of rankn over the ring of algebraic integersR _m in an imaginary quadratic field . It is proved that for anyn (in casem=11, there is one exceptionn=3) there exist indecomposable definite unimodular normal HermitianR ₁₅(R ₁₁)-lattices of rankn, and we exhibit representatives for each class. In the exceptional case there are no lattices with the desired properties. The method given in this paper can solve completely the problem of constructing indecomposable definite unimodular normal HermitianR _m -lattices of any rankn for eachm. Dedicated to the memory of Prof. Lee Hwa-Chung.     

Finite dimensional spectral inverse problem for a bundle of hermitian quadratic forms

The paper is devoted to recovering the coefficients of a pair of Hermitian quadratic forms c(x, x) and (x, x) in a special basis, in which the matrix of the form c(x, x) is tridiagonal and the matrix of the form m(x, x) is diagonal. The form c(x, x) is positive definite, and the form m(x, x) is nondegenerate, but is not positive difinite in contrast with a well-known case. The data of the inverse problem include the spectrum λ₁...,λ_n of the bundle II_λ and the set of numbers ρ₁...,ρ_n connected with the main normalized elements of the bundle. A procedure for solving the inverse problem is described. The characteristic conditions for λ₁...,λ_n; are found that provide its solution. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 33–36, 1990. Translated by T. N. Surkova.