Related papers: Counting Square Discriminants
We functorially identify similarity classes of line-bundle-valued quadratic forms on rank two vector bundles with isomorphism classes of pairs consisting of the degree zero and the degree one parts of the associated generalized Clifford…
We study the congruence of bitangent lines of an irreducible surface in the 3-dimensional projective space in arbitrary characteristic, with special attention to quartic surfaces with rational double points and, in particular, Kummer…
Dirichlet computed in some particular cases the number of equivalence classes of representations of a nonzero integer by a representative system for the integral binary quadratic forms of a given discriminant. We complete this computation.
We prove an upper bound for the length of an arithmetic progression represented by an irreducible integral binary quadratic form or a norm form, which depends only on the form and the progression's common difference. For quadratic forms,…
We evaluate the average of cubic and quartic Dirichlet character sums with the modulus going up to a size comparable to the length of the individual sums. This generalizes a result of Conrey, Farmer and Soundararajan on quadratic Dirichlet…
We establish sharp upper bounds on shifted moments of quadratic Dirichlet $L$-functions over function fields. As an application, we prove some bounds for moments of quadratic Dirichlet character sums over function fields.
Quadratic forms with embedded discriminants. Integral binary quadratic forms have multiple applications, for example in factorization or cryptography. The Nice family of cryptographic systems makes use of quadratic forms with different…
Let $f(\mathbf x)$ be a non-singular quadratic form with sufficiently many mixed terms and $t$ an integer. For a sequence of weights $\mathcal A$ we study the number of weighted solutions to $f(\mathbf x) = t$. In particular, we give…
We describe derivations of the Clifford algebra of a nondegenerate quadratic form on a countable dimensional vector space over an algebraically closed field of characteristic not equal to $2$. We also construct an algebraic automorphism of…
We consider the algebra of invariants of binary forms of degree 10 with complex coefficients, construct a system of parameters with degrees 2, 4, 6, 6, 8, 9, 10, 14 and find the 106 basic invariants.
The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this…
We establish effective versions of Oppenheim's conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed shift vectors and generic quadratic forms. When the shift is rational we prove a counting result which…
In this article, we investigate the conditional large values of quadratic Dirichlet character sums. We prove an Omega result for quadratic character sums under the assumption of the generalized Riemann hypothesis.
The continue fractions of quadratic surds are periodic, according to a theorem by Lagrange. Their periods may have differing types of symmetries. This work relates these types of symmetries to the symmetries of the classes of the…
We consider perturbations of Dirac type operators on complete, connected metric spaces equipped with a doubling measure. Under a suitable set of assumptions, we prove quadratic estimates for such operators and hence deduce that these…
A classical result of Dirichlet shows that certain elementary character sums compute class numbers of quadratic imaginary number fields. We obtain analogous relations between class numbers and a weighted character sum associated to a…
We construct the Dirichlet form associated with the dynamical $\Phi^4_3$ model obtained in [Hai14, CC13] and [MW16]. This Dirichlet form on cylinder functions is identified as a classical gradient bilinear form. As a consequence, this…
We study the singular series associated to a cubic form with integer coefficients. If the number of variables is at least $10$, we prove the absolute convergence (and hence positivity) under the assumption of Davenport's Geometric…
In this paper, we present an algorithm to compute a basis of the space of algebraic modular forms on the maximal order of the definite quaternion algebra of discriminant $2$, and provide a database of such bases. One of our motivations is…
The aim of this article is to study (additively) indecomposable algebraic integers $\mathcal O_K$ of biquadratic number fields $K$ and universal totally positive quadratic forms with coefficients in $\mathcal O_K$. There are given…