arXiv subjects

Kiseok Yeon

Publications and source records attributed to Kiseok Yeon.

10 recordsLinked to original sources

arXiv2026

On weighted forms in many variables

In this paper, we introduce several novel approaches utilizing the circle method to obtain the asymptotic formula for the number of integral points of bounded height lying on a hypersurface in a weighted projective space. Let $F(\mathbf{x} ; \mathbf{y})$ be a given weighted form of degree $d$ in variables $\mathbf{x} \in \mathbb{R}^{s_1}$ and $\mathbf{y} \in \mathbb{R}^{s_2}$, where variables $\mathbf{x}$ and $\mathbf{y}$ have weights $w_1$ and $w_2$ with $w_1 w_1 w_2$. Write $$ R_F(P):=\#\left\{(\mathbf{x} ; \mathbf{y}) \in \mathbb{Z}^{s_1+s_2}: F(\mathbf{x} ; \mathbf{y})=0,|\mathbf{x}| \leq P^{w_1 / d},|\mathbf{y}| \leq P^{w_2 / d}\right\} . $$ In particular, we show that whenever $$ s_1+s_2-\sigma_F>\left(1+\frac{w_2}{w_1}\right) \frac{d}{w_1} 2^{d / w_1}, $$ where $\sigma_F$ is the dimension of the affine singular locus of $F$, the quantity $R_F(P)$ has the expected asymptotic formula, that is $$ R_F(P)=c P^{s_1 w_1 / d+s_2 w_2 / d-1}+o\left(P^{s_1 w_1 / d+s_2 w_2 / d-1}\right), $$ where $c$ is the product of local densities. Furthermore, the constant $c$ is positive whenever $F(\mathbf{x} ; \mathbf{y})=0$ has a nonsingular solution over $\mathbb{R}$ and $\mathbb{Q}_p$ for every prime $p$. As a corollary, we verify the integral Hasse principle for the quasi-smooth hypersurface defined by $F(\mathbf{x} ; \mathbf{y})=0$ in a weighted projective space of sufficiently large dimensions.

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arXiv2026

On the density of rational lines on diagonal cubic hypersurfaces, II

In this paper, we establish the expected asymptotic formula for the number of rational lines on a diagonal cubic hypersurface in 18 or more variables, improving on recent work of the second author. This is achieved via a refined mean value estimate for minor arcs that non-trivially exploits a shifting variables argument in both underlying dimensions.

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arXiv2026

On the density of rational lines on diagonal cubic hypersurfaces

In this paper, we establish the asymptotic estimates for the rational lines on diagonal cubic hypersurfaces defined by $\sum_{i=1}^sc_ix^3_i=0$ with $c_i\in\mathbb{Z}\setminus \{0\},$ provided that $s\geq 19.$ This improves the previously known bound $s\geq 21$ required to obtain such asymptotic estimates. Our approach develops a multidimensional shifting variables argument together with a pruning argument, and exploits the recent progress on the Parsell-Vinogradov system.

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arXiv2025

The Hasse principle for random homogeneous polynomials in thin sets

Let $d$ and $n$ be natural numbers. Let $\nu_{d,n}: \mathbb{R}^n\rightarrow \mathbb{R}^{N}$ denote the Veronese embedding with $N=N_{n,d}:=\binom{n+d-1}{d}$, defined by listing all the monomials of degree $d$ in $n$ variables using the lexicographical ordering. Let $\langle \boldsymbol{a}, \nu_{d,n}(\boldsymbol{x})\rangle\in \mathbb{Z}[\boldsymbol{x}]$ be a homogeneous polynomial in $n$ variables of degree $d$ with integer coefficients $\boldsymbol{a}$, where $\langle\cdot,\cdot\rangle$ denotes the inner product. For a non-singular form $P\in \mathbb{Z}[\boldsymbol{x}]$ of degree $k\ (\leq d)$ in $N$ variables, consider a set of integer vectors $\boldsymbol{a}\in \mathbb{Z}^N$, defined by $$\mathfrak{A}(A;P)=\{\boldsymbol{a}\in \mathbb{Z}^N:\ P(\boldsymbol{a})=0,\ \|\boldsymbol{a}\|_{\infty}\leq A\}.$$ By handling a new lattice problem via the geometry of numbers, we confirm that whenever $n> 24d$ and $d\geq 17,$ the proportion of integer coefficients $\boldsymbol{a}\in \mathfrak{A}(A;P)$, whose associated equation $f_{\boldsymbol{a}}(\boldsymbol{x})=0$ satisfies the Hasse principle, converges to $1$ as $A\rightarrow\infty$. This improves on the recent work of the second author.

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arXiv2025

An extended Vinogradov's mean value theorem

In this paper, we provide novel mean value estimates for exponential sums related to the extended main conjecture of Vinogradov's mean value theorem, by developing the Hardy-Littlewood circle method together with a refined shifting variables argument. Let $d\geq 2$ be a natural number and $\boldsymbol{\alpha}=(\alpha_d,\ldots, \alpha_1)\in \mathbb{R}^d.$ Define the exponential sum \begin{equation*} f_d(\boldsymbol{\alpha};N):=\sum_{1 \leq n \leq N}e(\alpha_d n^d + \cdots+ \alpha_1 n). \end{equation*} For $p>0$, consider mean values of the exponential sums \begin{equation*} \mathcal{I}_{p,d}(u;N):=\int_{[0,1)\times [0,N^{-u})\times [0,1)^{d-2}}|f_d(\boldsymbol{\alpha};N)|^pd\boldsymbol{\alpha}, \end{equation*} where we wrote $d\boldsymbol{\alpha}=d\alpha_1 d\alpha_2\cdots d\alpha_{d-1}d\alpha_d.$ By making use of the aforementioned tools, we obtain the sharp upper bound for $\mathcal{I}_{p,d}(u;N)$, for $d=2,3$ and $0<u\leq 1$. Furthermore, for $d \geq 4$, we obtain analogous results depending on a small cap decoupling inequality for the moment curves in $\mathbb{R}^d.$

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arXiv2023

The local solubility for homogeneous polynomials with random coefficients over thin sets

Let $d$ and $n$ be natural numbers greater or equal to $2$. Let $\langle \boldsymbol{a}, \nu_{d,n}(\boldsymbol{x})\rangle\in \mathbb{Z}[\boldsymbol{x}]$ be a homogeneous polynomial in $n$ variables of degree $d$ with integer coefficients $\boldsymbol{a}$, where $\langle\cdot,\cdot\rangle$ denotes the inner product, and $\nu_{d,n}: \mathbb{R}^n\rightarrow \mathbb{R}^N$ denotes the Veronese embedding with $N=\binom{n+d-1}{d}$. Consider a variety $V_{\boldsymbol{a}}$ in $\mathbb{P}^{n-1}$, defined by $\langle \boldsymbol{a}, \nu_{d,n}(\boldsymbol{x})\rangle=0.$ In this paper, we examine a set of these varieties defined by $$\mathbb{V}^{P}_{d,n}(A)=\{ V_{\boldsymbol{a}}\subset \mathbb{P}^{n-1}|\ P(\boldsymbol{a})=0,\ \|\boldsymbol{a}\|_{\infty}\leq A\},$$ where $P\in \mathbb{Z}[\boldsymbol{x}]$ is a non-singular form in $N$ variables of degree $k$ with $2 \le k\leq C({n,d})$ for some constant $C({n,d})$ depending at most on $n$ and $d$. Suppose that $P(\boldsymbol{a})=0$ has a nontrivial integer solution. We confirm that the proportion of varieties $V_{\boldsymbol{a}}$ in $\mathbb{V}^{P}_{d,n}(A)$, which are everywhere locally soluble, converges to a constant $c_P$ as $A\rightarrow \infty.$ In particular, if there exists $\boldsymbol{b}\in \mathbb{Z}^N$ such that $P(\boldsymbol{b})=0$ and the variety $V_{\boldsymbol{b}}$ in $\mathbb{P}^{n-1}$ admits a smooth $\mathbb{Q}$-rational point, the constant $c_P$ is positive.

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arXiv2023

Small fractional parts of polynomials and mean values of exponential sums

Let $k_i\ (i=1,2,\ldots,t)$ be natural numbers with $k_1>k_2>\cdots>k_t>0$, $k_1\geq 2$ and $t<k_1.$ Given real numbers $α_{ji}\ (1\leq j\leq t,\ 1\leq i\leq s)$, we consider polynomials of the shape $$φ_i(x)=α_{1i}x^{k_1}+α_{2i}x^{k_2}+\cdots+α_{ti}x^{k_t},$$ and derive upper bounds for fractional parts of polynomials in the shape $$φ_1(x_1)+φ_2(x_2)+\cdots+φ_s(x_s),$$ by applying novel mean value estimates related to Vinogradov's mean value theorem. Our results improve on earlier Theorems of Baker (2017).

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arXiv2023

The Hasse principle for homogeneous polynomials with random coefficients over thin sets

In this paper, we investigate the solubility of homogeneous polynomial equations. The work of Browning, Le boudec, Sawin [3] shows that almost all homogeneous equations of degree $d\geq 4$ in $d+1$ or more variables satisfy the Hasse principle, and in particular that a positive portion possess a non-trivial integral solution. Our main result, when combined with our sequel joint work with H.Lee and S.Lee, shows that such a conclusion remains true even when the coefficients of homogeneous polynomials are constrained by a polynomial condition under a modest condition on the number of variables. To be precise, let $d$ and $n$ be natural numbers. Let $\nu_{d,n}: \mathbb{R}^n\rightarrow \mathbb{R}^N$ denote the Veronese embedding with $N=\binom{n+d-1}{d}$, defined by listing all the monomials of degree $d$ in $n$ variables using the lexicographical ordering. Let $\langle \boldsymbol{a}, \nu_{d,n}(\boldsymbol{x})\rangle\in \mathbb{Z}[\boldsymbol{x}]$ be a homogeneous polynomial in $n$ variables of degree $d$ with integer coefficients $\boldsymbol{a}$, where $\langle\cdot,\cdot\rangle$ denotes the inner product. For a non-singular form $P\in \mathbb{Z}[\boldsymbol{x}]$ in $N$ variables of degree $k\geq 2,$ consider a set of integer vectors $\boldsymbol{a}\in \mathbb{Z}^N$, defined by $$\mathfrak{A}(A;P)=\{\boldsymbol{a}\in \mathbb{Z}^N|\ P(\boldsymbol{a})=0,\ \|\boldsymbol{a}\|_{\infty}\leq A\}.$$ We confirm that when $d\geq 4$, $n$ is sufficiently large in terms of $d$, and $k\leq d,$ the proportion of integer vectors $\boldsymbol{a}\in \mathbb{Z}^N$ in $\mathfrak{A}(A;P)$, whose associated equations $\langle \boldsymbol{a}, \nu_{d,n}(\boldsymbol{x})\rangle=0$ satisfy the Hasse principle, converges to $1$ as $A\rightarrow \infty$. We make explicit a lower bound on $n$ guaranteeing this conclusion. In particular, we show that when $d\geq 14$ it suffices to take $n\geq 32d+17$.

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arXiv2022

Small fractional parts of binary forms

We obtain bounds on fractional parts of binary forms of the shape $$Ψ(x,y)=α_k x^k+α_l x^ly^{k-l}+α_{l-1}x^{l-1}y^{k-l+1}+\cdots+α_0 y^k$$ with $α_k,α_l,\ldots,α_0\in\mathbb{R}$ and $l\leq k-2.$ By exploiting recent progress on Vinogradov's mean value theorem and earlier work on exponential sums over smooth numbers, we derive estimates superior to those obtained hitherto for the best exponent $σ$, depending on $k$ and $l,$ such that \begin{equation*} \min_{\substack{0\leq x,y\leq X\\(x,y)\neq (0,0)}}\|Ψ(x,y)\|\leq X^{-σ+ε}.\end{equation*}

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arXiv2019

Newton polyhedrons and $L^p$ Sobolev estimations

The aim of this study is to provide a perspective to help understand the singular average operator over polynomial hypersurfaces. In particular, this perspective will provide brevity and the possibility of generalizing previous results dealing with the fundamental problem of determining the precise $L^p$ regularity enhancement for the average operators. In previous studies dealing with polynomials, the Newton polyhedron of a polynomial has been utilized to observe dominant monomials. In this study, we go further by discussing the involvements of other monomials in detail, by introducing several geometric values on the Newton polyhedron.

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