arXiv subjects

Daejun Kim

Publications and source records attributed to Daejun Kim.

17 recordsLinked to original sources

arXiv2025

Composition laws of binary quadratic forms and isolations of quadratic forms

A positive definite and integral quadratic form $f$ is called irrecoverable if there is a quadratic form $F$ such that it represents all proper subforms of $f$, whereas it does not represent $f$ itself. In this case, $F$ is called an isolation of $f$. In this article, we prove that there does not exist a binary isolation of any unary quadratic form. We also prove that there does not exist a ternary isolation of any binary quadratic form. Furthermore, if the form class group of a primitive binary quadratic form has no element of order $4$, then the discriminant of any quaternary isolation of it, if exists, is a square of an integer. The composition laws of primitive binary quadratic forms play an essential role in the proofs of the results.

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arXiv2025

Sums of squares of integers from residue classes

A subset $\mathcal{A}\subseteq\mathbb{Z}$ is called $s$-almost square universal if every sufficiently large positive integer can be written as a sum of at most $s$ squares of integers from $\mathcal{A}$. In this article, we study the minimal number $\mathrm{ASU}(\mathcal{A}_{d,m})$ with this property, where $\mathcal{A}_{d,m}$ denotes the residue class of $d$ modulo $m$, with $m\in\mathbb{N}$ and $d\in\mathbb{Z}$. We further prove that $\mathcal{A}_{d,m}$ is $s$-square universal for some $s\in\mathbb{N}$ if and only if $d \equiv \pm 1 \pmod{m}$, and determine the minimal such number $\mathrm{SU}(\mathcal{A}_{d,m})$ in these cases.

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arXiv2025

Zeta functions of quadratic lattices of a hyperbolic plane

In this paper, we study the Dirichlet series that enumerates proper equivalence classes of full-rank sublattices of a given quadratic lattice in a hyperbolic plane -- that is, a nondegenerate isotropic quadratic space of dimension $2$. We derive explicit formulas for the associated zeta functions and obtain a combinatorial way to compute them. Their analytic properties lead to the intriguing consequence that a large proportion of proper classes are one-lattice classes.

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arXiv2024

Sums of generalized polygonal numbers of almost prime "length"

In this paper, we consider sums of three generalized $m$-gonal numbers whose parameters are restricted to integers with a bounded number of prime divisors. With some restrictions on $m$ modulo $30$, we show that a density one set of integers is represented as such a sum, where the parameters are restricted to have at most 6361 prime factors. Moreover, if the squarefree part of $f_m(n)$ is sufficiently large, then $n$ is represented as such a sum, where $f_m(n)$ is a natural linear function in $n$.

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arXiv2024

Zeta functions enumerating subforms of quadratic forms

In this paper, we introduce and study the Dirichlet series enumerating (proper) equivalence classes of full rank subforms/sublattices of a given quadratic form/lattice, focusing on the positive definite binary case. We obtain formulas linking this Dirichlet series with Dirichlet series counting ideal classes of the imaginary quadratic field associated with the quadratic form. Utilizing the result, we provide explicit formulas of the Dirichlet series for several lattices, including square lattice and hexagonal lattice. Moreover, we investigate some analytic properties of this Dirichlet series.

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arXiv2024

Universality lifting from a general base field

Given a totally real number field $F$, we show that there are only finitely many totally real extensions of $K$ of a fixed degree that admit a universal quadratic form defined over $F$. We further obtain several explicit classification results in the case of relative quadratic extensions.

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arXiv2023

Isolations of the sum of two squares from its proper subforms

For a (positive definite and integral) quadratic form $f$, a quadratic form is said to be {\it an isolation of $f$ from its proper subforms} if it represents all proper subforms of $f$, but not $f$ itself. It was proved that the minimal rank of isolations of the square quadratic form $x^2$ is three, and there are exactly $15$ ternary diagonal isolations of $x^2$. Recently, it was proved that any quaternary quadratic form cannot be an isolation of the sum of two squares $I_2=x^2+y^2$, and there are quinary isolations of $I_2$. In this article, we prove that there are at most $231$ quinary isolations of $I_2$, which are listed in Table $1$. Moreover, we prove that $14$ quinary quadratic forms with dagger mark in Table $1$ are isolations of $I_2$.

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arXiv2023

Lifting problem for universal quadratic forms over totally real cubic number fields

Lifting problem for universal quadratic forms asks for totally real number fields $K$ that admit a positive definite quadratic form with coefficients in $\mathbb{Z}$ that is universal over the ring of integers of $K$. In this paper, we show that $K=\mathbb{Q}(\zeta_7+\zeta_7^{-1})$ is the only such totally real cubic field. Moreover, we show that there is no such biquadratic field.

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arXiv2022

Theta series of ternary quadratic lattice cosets

In this paper, we consider the decomposition of theta series for lattice cosets of ternary lattices. We show that the natural decomposition into an Eisenstein series, a unary theta function, and a cuspidal form which is orthogonal to unary theta functions correspond to the theta series for the genus, the deficiency of the theta series for the spinor genus from that of the genus, and the deficiency of the theta series for the class from that of the spinor genus, respectively. These three pieces are hence invariants of the genus, spinor genus, and class, respectively, extending known results for lattices and verifying a conjecture of the first author and Haensch. We furthermore extend the definition of $p$-neighbors to include lattice cosets and construct an algorithm to compute respresentatives for the classes in the genus or spinor genus via the $p$-neighborhoods.

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arXiv2022

Explicit Class number formulas for Siegel--Weil averages of ternary quadratic forms

In this paper, we investigate the interplay between positive-definite integral ternary quadratic forms and class numbers. We generalize a result of Jones relating the theta function for the genus of a quadratic form to the Hurwitz class numbers, obtaining an asymptotic formula (with a main term and error term away from finitely many bad square classes $t_j\mathbb{Z}^2$) relating the number of lattices points in a quadratic space of a given norm with a sum of class numbers related to that norm and the squarefree part of the discriminant of the quadratic form on this lattice.

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arXiv2022

Primitively universal quaternary quadratic forms

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 say $f$ is $\textit{primitively almost universal}$ if it represents almost all positive integers primitively. Conway and Schneeberger proved (see [1]) that there are exactly $204$ equivalence classes of universal quaternary quadratic forms. Recently, Earnest and Gunawardana proved in [4] that among $204$ equivalence classes of universal quaternary quadratic forms, there are exactly $152$ equivalence classes of primitively almost universal quaternary quadratic forms. In this article, we prove that there are exactly $107$ equivalence classes of primitively universal quaternary quadratic forms. We also determine the set of all positive integers that are not primitively represented by each of the remaining $152-107=45$ equivalence classes of primitively almost universal quaternary quadratic forms.

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arXiv2020

The pentagonal theorem of sixty-three and generalizations of Cauchy's lemma

In this article, we study the representability of integers as sums of pentagonal numbers, where a pentagonal number is an integer of the form $P_5(x)=\frac{3x^2-x}{2}$ for some non-negative integer $x$. In particular, we prove the "pentagonal theorem of $63$", which states that a sum of pentagonal numbers represents every non-negative integer if and only if it represents the integers $1$, $2$, $3$, $4$, $6$, $7$, $8$, $9$, $11$, $13$, $14$, $17$, $18$, $19$, $23$, $28$, $31$, $33$, $34$, $39$, $42$, and $63$. We also introduce a method to obtain a generalized version of Cauchy's lemma using representations of binary integral quadratic forms by quaternary quadratic forms, which plays a crucial role in proving the results.

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arXiv2020

Prime-universal diagonal quadratic forms

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.

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arXiv2020

Weighted sums of generalized polygonal numbers with coefficients 1 or 2

In this article, we consider weighted sums of generalized polygonal numbers with coefficients $1$ or $2$. We show that for any $m\ge10$, those weighted sums of generalized $m$-gonal numbers represent every non-negative integers if they only represent $1$, $m-4$, and $m-2$. Furthermore, we study representations of sums of four generalized polygonal numbers with coefficients $1$ or $2$.

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arXiv2019

Representations of finite number of quadratic forms with same rank

Let $m, n$ be positive integers with $m\le n$. Let $\kappa(m,n)$ be the largest integer $k$ such that for any (positive definite and integral) quadratic forms $f_1,\ldots,f_k$ of rank $m$, there exists a quadratic form of rank $n$ that represents $f_i$ for any $i$ with $1\le i \le k$. In this article, we determine the number $\kappa(m,n)$ for any integer $m$ with $1\le m\le 8$, except for the cases when $(m,n)=(3,5)$ and $(4,6)$. In the exceptional cases, it will be proved that $1\le \kappa(3,5), \ \kappa(4,6)\le 2$. We also discuss some related topics.

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arXiv2019

A sum of three nonunit squares of integers

We say a positive integer is a sum of three nonunit squares if it is a sum of three squares of integers other than one. In this article, we find all integers which are sums of three nonunit squares assuming that the Generalized Riemann Hypothesis(GRH) holds. As applications, we find all integers, under the GRH only when $k=3$, which are sums of $k$ nonzero triangular numbers, sums of $k$ nonzero generalized pentagonal numbers, and sums of $k$ nonzero generalized octagonal numbers, respectively for any integer $k\ge 3$.

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arXiv2019

On a quadratic Waring's problem with congruence conditions

For each positive integer $n$, let $g_\Delta(n)$ be the smallest positive integer $g$ such that every complete quadratic polynomial in $n$ variables which can be represented by a sum of odd squares is represented by a sum of at most $g$ odd squares. In this paper, we analyze $g_\Delta(n)$ by studying representations of integral quadratic forms by sums of squares with certain congruence condition. We prove that the growth of $g_\Delta(n)$ is at most an exponential of $\sqrt{n}$, which is the same as the best known upper bound on the $g$-invariants of the original quadratic Waring's problem. We also determine the exact value of $g_\Delta(n)$ for each positive integer less than or equal to $4$.

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