Related papers: On ternary quadratic forms over the rational numbe…

An integral quadratic polynomial (with positive definite quadratic part) is called almost universal if it represents all but finitely many positive integers. In this paper, we introduce the conductor of a quadratic polynomial, and give an…

We establish a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms using an analytic number theory approach. The statements come with power gains and in some cases are essentially optimal

We re-examine previous constructions of infinite binary words containing few distinct squares with the goal of finding the "simplest", in a certain sense. We exhibit several new constructions. Rather than using tedious case-based arguments…

We show that if a universal quadratic form exists over an infinite degree, totally real extension of the field of rationals $\mathbb{Q}$, then the set of totally positive integers in the extension does not have the Northcott property. In…

Lagrange's four-square theorem states that every natural number $n$ can be represented as the sum of four integer squares: $n=x_1^2+x_2^2+x_3^2+x_4^2$. Ramanujan generalized Lagrange's result by providing, up to equivalence, all $54$…

Lagrange's Four Squares Theorem states that any positive integer can be expressed as the sum of four integer squares. We investigate the analogous question over Quaternion rings, focusing on squares of elements of Quaternion rings with…

For a totally positive definite quadratic form over the ring of integers of a totally real number field $K$, we show that there are only finitely many totally real field extensions of $K$ of a fixed degree over which the form is universal…

We derive an upper bound for the least number of variables needed to guarantee that a system of t quadratic forms (t>=2) over a field F has a nontrivial zero. In particular, if F is a local field, then 2t^2+3 variables insure the existence…

Let $A$ be a definite quaternion algebra over $\mathbb Q$, with discriminant $D_A$, and $O$ a maximal order of $A$. We show that the minimum of the positive definite hamiltonian binary forms over $O$ with discrimiminant $-1$ is…

In this paper, we find a basis for the space of modular forms of weight $2$ on $\Gamma_1(48)$. We use this basis to find formulas for the number of representations of a positive integer $n$ by certain quaternary quadratic forms of the form…

We characterize the generating function of the number of representations described in the title in terms of the theory of modular forms. Appealing to this characterization we obtain explicit formulas for the representation numbers as…

For any fixed nonzero integer $h$, we show that a positive proportion of integral binary quartic forms $F$ do locally everywhere represent $h$, but do not globally represent $h$. We order classes of integral binary quartic forms by the two…

For any positive integer M we show that there are infinitely many real quadratic fields that do not admit M-ary universal quadratic forms (without any restriction on the parity of their cross coefficients).

In this paper, we are interested in the interplay between integral ternary quadratic forms and class numbers. This is partially motivated by a question of Petersson.

Positive and negative quadratic forms are well known and widely used. They are multivariate homogeneous polynomials of degree two taking positive or negative values respectively for any values of their arguments not all zero. In the present…

In this paper, we find formulas for the number of representations of certain diagonal octonary quadratic forms with coefficients $1,2,3,4$ and $6$. We obtain these formulas by constructing explicit bases of the space of modular forms of…

Let $Q(x_1, \cdots,x_n)$ be a real indefinite quadratic form of the type $(r,s)$, $n=r+s$, signature $\sigma=r-s$ and determinant $D\neq 0$. Let $\Gamma_{r,n-r}$ denote the infimum of all numbers $\Gamma$ such that for any real numbers…

Using the circle method, we show that for a fixed positive definite integral quadratic form $A$, the expected asymptotic formula for the number of representations of a positive definite integral quadratic form $B$ by $A$ holds true,…

Quantifier elimination of positive semidefinite cyclic ternary quartic forms is studied in this paper. We solve the problem by the theory of complete discrimination systems, function \RealTriangularize in Maple15 and the so-called…

Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by $x$ over which there exist elliptic curves with good reduction everywhere and rational $j$-invariant is $\gg…