Related papers: On $n$-universal quadratic forms over dyadic local…

In this paper we will give a unified proof of several results on the sovability of systems of certain equations over finite fields, which were recently obtained by Fourier analytic methods. Roughly speaking, we show that almost all systems…

Formulating quantum integrability for nonultralocal models (NM) parallel to the familiar approach of inverse scattering method is a long standing problem. After reviewing our result regarding algebraic structures of ultralocal models, we…

Berry and Tabor conjectured in 1977 that spectra of generic integrable quantum systems have the same local statistics as a Poisson point process. We verify their conjecture in the case of the two-point spectral density for a quantum…

We settle an old question about the existence of certain "sums-of-squares" formulas over a field F (which are the simplest examples of composition formulas for quadratic forms). A classical theorem says that if such a formula exists over a…

A (positive definite and integral) quadratic form $f$ is said to be $\textit{universal}$ if it represents all positive integers, and is said to be $\textit{primitively universal}$ if it represents all positive integers primitively. We also…

In two dimensions, Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fibre version of Gallagher's theorem, sharpening and making unconditional a…

Let $K$ be the fraction field of a 2-dimensional, henselian, excellent local domain with finite residue field $k$. When the characteristic of $k$ is not 2, we prove that every quadratic form of rank $\ge 9$ is isotropic over $K$ using…

We prove the Hasse principle and weak approximation for varieties defined over number fields by the nonsingular intersection of pairs of quadratic forms in 8 variables. The argument develops work of Colliot-Thelene, Sansuc and…

In this paper, we give an equivalent condition for an abelian variety over a finite field to have multiplication by a quaternion algebra over a number field. We prove the result by combining Tate's classification of the endomorphism…

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.

Inspired by a paper of Salberger we give a new proof of Manin's conjecture for toric varieties over imaginary quadratic number fields by means of universal torsor parameterizations and elementary lattice point counting.

Genus Theory is a classical feature of integral binary quadratic forms. Using the author's generalization of the well-known correspondence between quadratic form classes and ideal classes of quadratic algebras, we extend it to the case when…

Let k be a global field of characteristic not 2. We prove a local-global principle for the existence of self-dual normal bases, and more generally for the isomorphism of G-trace forms, of G-Galois algebras over k.

In this work, we generalize Sacks-Uhlenbeck's existence result for harmonic spheres, constructing for $n \ge 2$, regular, non-trivial, $n$-harmonic $n$-spheres into suitable target manifolds. We obtain an infinite family of new…

Metric independent $\sigma$ models are constructed. These are field theories which generalise the membrane idea to situations where the target space has fewer dimensions than the base manifold. Instead of reparametrisation invariance of the…

A version of the Davis-Kahan Tan $2\Theta$ theorem [SIAM J. Numer. Anal. \textbf{7} (1970), 1 -- 46] for not necessarily semibounded linear operators defined by quadratic forms is proven. This theorem generalizes a recent result by…

We use the theory of the quantum group $U_q(gl(2,\RR))$ in order to develop a quantum theory of invariants and show a decomposition of invariants into a Gordan-Capelli series. Higher binary forms are introduced on the basis of braided…

A positive quadratic form is $(k,\ell)$-universal if it represents all the numbers $kx+\ell$ where $x$ is a non-negative integer, and almost $(k,\ell)$-universal if it represents all but finitely many of them. We prove that for any $k,\ell$…

In addition to rather complicated general methods it is interesting and valuable to develop fast efficient methods for calculating generators of power integral bases in special types of number fields. We consider sextic fields containing a…

In this paper the author considers a particular type of polynomials with integer coefficients, consisting of a perfect power and two norm forms of abelian number fields with coprime discriminants. It is shown that such a polynomial…