T. D. Browning
Let Q be a non-singular diagonal quadratic form in at least four variables. We provide upper bounds for the number of integer solutions to the equation Q=0, which lie in a box with sides of length 2B, as B tends to infinity. The estimates obtained are completely uniform in the coefficients of the form, and become sharper as they grow larger in modulus.
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A (positive definite and integral) quadratic form is said to be $\textit{prime-universal}$ if it represents all primes. Recently, Doyle and Williams in [2] classified all prime-universal diagonal ternary quadratic forms, and all prime-universal diagonal quaternary quadratic forms under two conjectures proposed by themselves. In this article, we classify all prime-universal diagonal quadratic forms regardless of ranks. Furthermore, we prove, so called, $67$-Theorem for a diagonal quadratic form to be prime-universal.
A positive-definite integral quadratic form is called regular if it represents every positive integer which is locally represented. In this article, we classify all regular diagonal quadratic forms of rank greater than 3.
A (positive definite primitive integral) quadratic form is called odd-regular if it represents every odd positive integer which is locally represented. In this paper, we show that there are at most 147 diagonal odd-regular ternary quadratic forms and prove the odd-regularities of all but 6 candidates.
A (positive definite integral) quadratic form is called almost 2-universal if it represents all (positive definite integral) binary quadratic forms except those in only finitely many equivalence classes. Oh [7] determined all almost 2-universal quinary diagonal quadratic forms remaining three as candidates. In this article, we prove that those three candidates are indeed almost 2-universal.
We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel's mass formula) can be used to compute the representation numbers of certain ternary quadratic forms.
Let F be a perfect field. Then the diagonal quadratic form $a_iX_i^2$ over $F$ is universal over $M_2(F)$ if and only if atleast two of the $a_i$ are non-zero.
For suitable pairs of diagonal quadratic forms in 8 variables we use the circle method to investigate the density of simultaneous integer solutions and relate this to the problem of estimating linear correlations among sums of two squares.
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
Employing Brüdern's and Wooley's new complification method, we establish an asymptotic Hasse principle for the number of solutions to a system of r_3 cubic and r_2 quadratic diagonal forms, when the number of cubic equations is at least double the number of quadratic forms. The number of variables required is lower than in earlier work. In particular, we achieve essentially square root cancellation for systems of one quadratic and arbitrarily many cubic equations.
This paper investigates the upper bound of the number of integer (natural) solutions of inhomogeneous algebraic Diophantine diagonal equations with integer coefficients without a free member via the circle method of Hardy and Littlewood. Author found the upper bound of the number of natural solutions of inhomogeneous algebraic Diophantine diagonal equations with explicit variable. He developed a method in the paper, which allows you to perform the low estimate of the number of natural (integer) solutions of algebraic Diophantine equation with integer coefficients. Author obtained a lower estim…
The author showed that any homogeneous algebraic Diophantine equation of the second order can be converted to a diagonal form using an integer non-orthogonal transformation maintaining asymptotic behavior of the number of its integer solutions. In this paper, we consider the transformation to the diagonal form of a wider class of algebraic second-order Diophantine equations, and also we consider the conditions for converting higher order algebraic Diophantine equations to this form. The author found an asymptotic estimate for the number of integer solutions of the diagonal Thue equation of odd…
The paper presents a new generalized integer orthogonal transformation which consists of a well known orthogonal transform followed by stretching the basis vectors maintaining the asymptotic behavior of the number of integer solutions for algebraic Diophantine equation. The author shows the properties of this transformation and he receives the algorithm for finding the matrix elements of a generalized integer orthogonal transformation for algebraic Diophantine equation of the second order to diagonal form. The article includes examples illustrating the reduction of algebraic equations of the s…
In this paper, we investigate the conditions under which a diagonal quadratic form $\sum_{i=1}^{m}a_i X_i^2$ represents every $n \times n$ integral matrix, where $a_i$ ($1 \leq i \leq m$) are integers. For $n=2$, we give a necessary and sufficient condition. Also we give some sufficient conditions for each $n \geq 2$ where $a_i$ ($1 \leq i \leq m$) are pairwisely coprime.
Covariants and simultaneous diagonalization of pairs of ternary quadratic forms, and binary quartic forms
本文建立了二元四次型与三元二次型对协变间的对应，并给出了任意特征零域上二次型对的标准对角化方法及有理数域上可对角化的精确判据。
Let $X$ be a sufficiently large positive integer. We prove that one may choose a subset $S$ of primes with cardinality $O(\log X)$, such that a positive proportion of integers less than $X$ can be represented by $x^2 + p y^2$ for at least one of $p \in S$.
The main goal of this note is to establish the limits of L. Zhao's techniques for counting solutions to quadratic forms in prime variables. Zhao considered forms with rank at least 9, and showed that these equations have solutions in primes provided there are no local obstructions. We consider in detail the degenerate cases of off-diagonal rank 1 and 2, and improve the rank lower bounds to at least 6 and at least 8 respectively. These results complement a recent breakthrough of Green on the non-degenerate rank 8 case.
Rational solutions of pairs of diagonal equations, one cubic and one quadratic
本文通过获得指数和 $αx^3+βx^2$ 的 32/3 阶矩的最优估计，在局部可解性假设下证明变量数超过 10 时一对三次和二次对角方程存在非平凡整数解。
We develop an algebraic method of studying of Diophantine quadratic equations in three variables over the ring of Gaussian integers.
Author developed a method in the paper, which, unlike the circle method of Hardy and Littlewood (CM), allows you to perform a lower estimate for the number of natural (integer) solutions of algebraic Diophantine equation with integer coefficients. It was found the lower estimate of the number of natural solutions to various types of homogeneous algebraic Diophantine equations with integer coefficients diagonal form with any number of variables using this method. Author obtained upper bound of the number of the natural solutions (using CM) of one type of homogeneous Diophantine equation for val…
Following Bhargava and Hanke's celebrated 290-theorem, we prove a universality theorem for all positive-definite integer-valued quadratic forms that represent all positive integers coprime to $3$. In particular, if a positive-definite quadratic form represents all positive integers coprime to $3$ and $\leq 290$, then it represents all positive integers coprime to $3$. We use similar methods to those used by Rouse to prove (assuming GRH) that a positive-definite quadratic form representing every odd integer between $1$ and $451$ represents all positive odd integers.
For any $\varepsilon > 0$ we derive effective estimates for the size of a non-zero integral point $m \in \mathbb{Z}^d \setminus \{0\}$ solving the Diophantine inequality $\lvert Q[m] \rvert < \varepsilon$, where $Q[m] = q_1 m_1^2 + \ldots + q_d m_d^2$ denotes a non-singular indefinite diagonal quadratic form in $d \geq 5$ variables. In order to prove our quantitative variant of the Oppenheim conjecture, we extend an approach developed by Birch and Davenport [BD58b] to higher dimensions combined with a theorem of Schlickewei [Sch85]. The result obtained is an optimal extension of Schlickewei's …
In 1997, Kaplansky conjectured that if two positive definite ternary quadratic forms with integer coefficients have perfectly identical integral representations, then they are isometric, both regular, or included either of two families of ternary quadratic forms. In this article, we prove the existence of pairs of ternary quadratic forms representing same integers which are not in the Kaplansky's list.
We give an elementary approach to studying whether rings of $S$-integers in complex quadratic fields are Euclidean with respect to the $S$-norm.
Representing rational integers by generalized quadratic forms over quadratic fields
研究二次域上值域为有理整数的广义二次型，证明实二次域中存在正定二元广义二次型可表示所有正整数，并证明仅有有限多个二次域存在三元广义二次型具有此类性质。
Suppose $k$ is a positive integer. In this work, we establish formulas for for the number of representations of integers by the quadratic forms $$ x_{1}^{2}+\cdots+x_{k}^{2}+l\left(x_{k+1}^{2}+\cdots+x_{2k}^{2}\right) $$ for $l\in\{2,4\}$.
In this paper we consider certain quaternary quadratic forms and octonary quadratic forms and by using the theory of modular forms, we find formulae for the number of representations of a positive integer by these quadratic forms.
We apply the circle method with a Gaussian weight to obtain an asymptotic formula for the density of representations of non-zero integers by non-singular quadratic forms in at least four variables.
In this note we show that for a given irreducible binary quadratic form $f(x,y)$ with integer coefficients, whenever we have $f(x,y) = f(u,v)$ for integers $x,y,u,v$, there exists a rational automorphism of $f$ which sends $(x,y)$ to $(u,v)$.
Building on the classification of all characteristic polynomials of integer symmetric matrices having small span (span less than 4), we obtain a classification of small-span polynomials that are the characteristic polynomial of a Hermitian matrix over some quadratic integer ring. Taking quadratic integer rings as our base, we obtain as characteristic polynomials some low-degree small-span polynomials that are not the characteristic (or minimal) polynomial of any integer symmetric matrix.
We use ideas from our previous work to obtain some theorems that will allow us to obtain the integer solution of a quadratic polynomial in two variables that represents a natural number
We classify all cyclotomic matrices over real quadratic integer rings and we show that this classification is the same as classifying cyclotomic matrices over the compositum all real quadratic integer rings. Moreover, we enumerate a related class of symmetric matrices; those matrices whose eigenvalues are contained inside the interval [-2,2] but whose characteristic polynomials are not in Z[x].
Sums of squares of integer-multiple of an integral element on real bi-quadratic fields
本文在实双二次域中构造全正代数整数，给出其正整数倍不能表示为整数平方和的必要条件，并分两类整数以表格示例验证。
Let Q be a non-singular quadratic form with integer coefficients. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q=0. When Q is positive definite we provide improved upper bounds for the least positive integer k such that the equation Q=k is insoluble in integers, despite being soluble modulo every prime power.
We give an infinite family of polynomials that have roots modulo every positive integer but fail to have rational roots. Each polynomial in this family is made up of monic quadratic factors that do not have linear term.
We investigate the existence of representations of every large positive integer as a sum of $k$-th powers of integers represented as certain diagonal forms. In particular, we consider a family of diagonal forms and discuss the problem of giving a uniform upper bound over the family for the number of variables needed to have such representations.
Let $K=\mathbb Q(\sqrt D)$ be a real quadratic field. We consider the additive semigroup $\mathcal O_K^+(+)$ of totally positive integers in $K$ and determine its generators (indecomposable integers) and relations; they can be nicely described in terms of the periodic continued fraction for $\sqrt D$. We also characterize all uniquely decomposable integers in $K$ and estimate their norms. Using these results, we prove that the semigroup $\mathcal O_K^+(+)$ completely determines the real quadratic field $K$.
Rational integers as sums of units -- the quadratic case
本文研究了二次数域中自然数作为k个单位和的渐进行为，解决了Jarden和Narkiewicz在2007年提出的问题。
We generalise remarks of Euler and of Perron by explaining how to detail all quadratic irrational integers for which the symmetric part of the period of their continued fraction expansion commences with prescribed partial quotients. The function field case is particularly striking.
We prove that for any countable set $A$ of real numbers, the set of binary indefinite quadratic forms $Q$ such that the closure of $Q(\mathbb{Z}^2)$ is disjoint from $A$ has full Hausdorff dimension.
The system of equations \[ u_1p_1^2 + \ldots + u_sp_s^2 = 0 \] \[ v_1p_1^3 + \ldots + v_sp_s^3 = 0 \] has prime solutions $(p_1, \ldots, p_s)$ for $s \geq 12$, assuming that the system has solutions modulo each prime $p$. This is proved via the Hardy-Littlewood circle method, building on Wooley's work on the corresponding system over the integers and recent results on Vinogradov's mean value theorem. Additionally, a set of sufficient conditions for local solvability is given: If both equations are solvable modulo 2, the quadratic equation is solvable modulo 3, and for each prime $p$ at least 7…
In this paper we obtain cyclic pentagons and hexagons with rational sides, diagonals and area all of which are expressed in terms of rational functions of several arbitrary rational parameters. On suitable scaling, we obtain cyclic pentagons and hexagons whose sides, diagonals and area are all given by integers.
We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n=x^2+Ny^2 for a squarefree integer N.
An integral quadratic polynomial is called regular if it represents every integer that is represented by the polynomial itself over the reals and over the $p$-adic integers for every prime $p$. It is called complete if it is of the form $Q({\mathbf x} + {\mathbf v})$, where $Q$ is an integral quadratic form in the variables ${\mathbf x} = (x_1, \ldots, x_n)$ and ${\mathbf v}$ is a vector in ${\mathbb Q}^n$. Its conductor is defined to be the smallest positive integer $c$ such that $c{\mathbf v} \in {\mathbb Z}^n$. We prove that for a fixed positive integer $c$, there are only finitely many equ…
On the sparsity of non-diagonalisable integer matrices and matrices with a given discriminant
研究整数矩阵集合中特征多项式具有固定判别式d的矩阵数量，对d=0的情况改进了之前的结果，对d≠0的情况提供了上界估计。
It is shown that when a real quadratic integer $ξ$ of fixed norm $μ$ is considered, the fundamental unit $\varepsilon_d$ of the field $\mathbb{Q}(ξ) = \mathbb{Q}(\sqrt{d})$ satisfies $\log \varepsilon_d \gg (\log d)^2$ almost always. An easy construction of a more general set containing all the radicands $d$ of such fields is given via quadratic sequences, and the efficiency of this substitution is estimated explicitly. When $μ= -1$, the construction gives all $d$'s for which the negative Pell's equation $X^2 - d Y^2 = -1$ (or more generally $X^2 - D Y^2 = -4$) is soluble. When $μ$ is a prime,…
In this article we show that the form $x^2 + iy^2 + z^2 + iw^2$ represents all gaussian integers. The main tools used in this proof are Fermat's little theorem (over finite field extensions), the Mordell-Niven theorem (representation of some gaussians), and the generalized Euler-identity over finite field extensions.
We identify and analyse obstructions to factorisation of integer matrices into products $N^T N$ or $N^2$ of matrices with rational or integer entries. The obstructions arise as quadratic forms with integer coefficients and raise the question of the discrete range of such forms. They are obtained by considering matrix decompositions over a superalgebra. We further obtain a formula for the determinant of a square matrix in terms of adjugates of these matrix decompositions, as well as identifying a $\it co-Latin$ symmetry space.
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 examples.
We study totally positive, additively indecomposable integers in a real quadratic field $\mathbb Q(\sqrt D)$. We estimate the size of the norm of an indecomposable integer by expressing it as a power series in $u_i^{-1}$, where $\sqrt D$ has the periodic continued fraction expansion $[u_0, u_1, u_2, \dots, u_{s-1}, 2u_0, u_1, u_2, \dots]$. This enables us to disprove a conjecture of Jang-Kim [JK] concerning the maximal size of the norm of an indecomposable integer.
We prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent "almost all" pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity condition. In particular such forms simultaneously attain prime values if the obvious local conditions hold. The proof uses the circle method, and in particular pioneers a two-dimensional version of a Kloosterman refinement.
Counting solutions to quadratic forms in eight prime variables of off-diagonal rank $3$
本文延续前人对素变量二次型解的计数研究，针对非对角秩为3的八个素变量非退化二次型，开展解的数量计数工作。
Cubic residuacity of real quadratic integers
针对带三次范数的实二次整数，本文确定相关型类群中表示对应素数的所有类，证明了2025年相关论文提出的一个特例猜想。
In 1957 N.C. Ankeny provided a new proof of the three squares theorem using geometry of numbers. This paper generalizes Ankeny's technique, proving exactly which integers are represented by $x^2 + 2y^2 + 2z^2$ and $x^2 + y^2 + 2z^2$ as well as proving sufficient conditions for an integer to be represented by $x^2+y^2+3z^2$ and $x^2 + y^2 + 7z^2$.
Let $R=\mathcal{O}_{\Q(\sqrt{d})}$ for $d<0$, squarefree, $d\neq -1,-3$. We prove Lehmer's conjecture for associated reciprocal polynomials of $R$-matrices; that is, any noncyclotomic $R$-matrix has Mahler measure at least $λ_0=1.176...$.
The least quadratic residue and integers represented by quadratic forms
该研究针对模n的最小非平凡既约二次剩余ℓ(n)，从n的大小与素因子数维度建立近乎最优界，构造出ℓ(n)异常大的模，并将结果应用于推导有界判别式二元二次型表示不超过N的正整数的速率界。
We consider numeration systems with base $β$ and $-β$, for quadratic Pisot numbers $β$ and focus on comparing the combinatorial structure of the sets $\Z_β$ and $\Z_{-β}$ of numbers with integer expansion in base $β$, resp. $-β$. Our main result is the comparison of languages of infinite words $u_β$ and $u_{-β}$ coding the ordering of distances between consecutive $β$- and $(-β)$-integers. It turns out that for a class of roots $β$ of $x^2-mx-m$, the languages coincide, while for other quadratic Pisot numbers the language of $u_β$ can be identified only with the language of a morphic image of …
By considering the norm of elements in the ring of integers in $\mathbb{Q}(\sqrt{-a})$, we give an algebraic approach to count the number of integral solutions of diophantine equations of the form $x^2+ay^2=n$ where $a$ is a Heegner number or $a=27$.
Criterion set for diagonal forms of higher degree
本文研究全实代数数域上的高次对角型，证明了存在唯一且极小的有限判据集合来判定其全向性，并建立了渐近局部-全局原理及局部表示理论。
A number field $k$ admits a binary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is bigger than one. In this case, there are only finitely many classes of such binary integral quadratic forms over $k$. A number field $k$ admits a ternary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is even. In this case, there are infinitely many classes of such ternary integral quadratic forms over $k$. An integral quadratic form over a number field $k$ with more tha…
In this paper, we consider the following diagonal quadratic forms \begin{equation*} a_1x_1^2 + a_2x_2^2 + \cdots + a_{\ell}x_{\ell}^2, \end{equation*} where $\ell\ge 5$ is an odd integer and $a_i\ge 1$ are integers. By using the extended Shimura correspondence, we obtain explicit formulas for the number of representations of $|D|n^2$ by the above type of quadratic forms, where $D$ is either a square-free integer or a fundamental discriminant such that $(-1)^{(\ell-1)/2}D > 0$. We demonstrate our method with many examples, in particular, we obtain all the formulas (when $\ell =5$) obtained in t…
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