Related papers: The divisor problem for binary cubic forms

It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential…

We provide the solution to the normal ordering problem for powers and exponentials of two classes of operators. The first one consists of boson strings and more generally homogeneous polynomials, while the second one treats operators linear…

We study the equidistribution of integers of the form $n= x_1^2 + \cdots + x_d^2$ under the arithmetic constraints given by $(\mathbb{Z}/p\mathbb{Z})^d$. The first step in addressing this problem is to construct modular forms whose Fourier…

Asymptotic formulae for Titchmarsh-type divisor sums are obtained with strong error terms that are uniform in the shift parameter. This applies to more general arithmetic functions such as sums of two squares, improving the error term in…

We present a complete computational classification of the combinatorial types of hyperplane sections, or slices, of the regular cube up to dimension six. For each dimension, we determine the exact number of distinct combinatorial types.…

We establish power saving asymptotics for the sum of the divisor function along a binary quartic form, improving on work of Daniel. The proof involves an application of a recent two dimensional delta method due to Li, Rydin-Myerson, and…

We describe the structure of bimodules (over finite dimensional algebras) which have the property that the functor of tensoring with such a bimodule sends any module to a projective module. The main result is that all such bimodules are…

For a binary quadratic form $Q$, we consider the action of $\mathrm{SO}_Q$ on a two-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and…

Estimating averages of Dirichlet convolutions $1 \ast \chi$, for some real Dirichlet character $\chi$ of fixed modulus, over the sparse set of values of binary forms defined over $\mathbb{Z}$ has been the focus of extensive investigations…

We construct a normal form for the walled Brauer algebra, together with the reduction algorithm. We apply normal form to calculate the numbers of monomials in generators with minimal length. We further utilize normal form to give explicit…

We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully…

Let $k$ be a number field and $O$ the ring of integers. In the previous paper [T06] we study the Dirichlet series counting discriminants of cubic algebras of $O$ and derive some density theorems on distributions of the discriminants by…

We study the distribution of divisors of Euler's totient function and Carmichael's function. In particular, we estimate how often the values of these functions have "dense" divisors.

In this paper we introduce a binomial ideal derived from a binary linear code. We present some applications of a Gr\"obner basis of this ideal with respect to a total degree ordering. In the first application we give a decoding method for…

We classify all cubic extensions of any field of arbitrary characteristic, up to isomorphism, via an explicit construction involving three fundamental types of cubic forms. We deduce a classification of any Galois cubic extension of a…

We consider the distribution of the largest prime divisor of the integers in the interval $[2,x]$, and investigate in particular the mode of this distribution, the prime number(s) which show up most often in this list. In addition to giving…

In this paper we propose an algorithm for the numerical solution of arbitrary differential equations of fractional order. The algorithm is obtained by using the following decomposition of the differential equation into a system of…

For the system of second order quasilinear parabolic equations the problem of reducing them to the equations of diffusion type is considered. In non-degenerate case an effective algorithm for solving this problem is suggested.

We investigate a version of Waring's Problem over quaternion rings, focusing on cubes in quaternion rings with integer coefficients. We determine the global upper and lower bounds for the number of cubes necessary to represent all such…

We study the behavior of volumes of divisors in a family. We show that the volume of a divisor on the generic fiber equals the infimum of its volumes on fibers over any dense subset of the base. As an application, we show that the volume…