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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., Comment: All comments are welcome!

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 $n\geq 32d+17$., Comment: 81 pages, all comments are welcome!

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

We obtain bounds on fractional parts of binary forms of the shape $$\Psi(x,y)=\alpha_k x^k+\alpha_l x^ly^{k-l}+\alpha_{l-1}x^{l-1}y^{k-l+1}+\cdots+\alpha_0 y^k$$ with $\alpha_k,\alpha_l,\ldots,\alpha_0\in\mathbb{R}$ 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 $\sigma$, depending on $k$ and $l,$ such that \begin{equation*} \min_{\substack{0\leq x,y\leq X\\(x,y)\neq (0,0)}}\|\Psi(x,y)\|\leq X^{-\sigma+\epsilon}.\end{equation*}, Comment: 30 pages, all comments welcome

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.