arXiv subjects
Publications and source records attributed to Shuntaro Yamagishi.
Let $F \in \mathbb{Z}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d \geq 4$ which defines a smooth hypersurface in $\mathbb{P}^{n-1}_{\mathbb{C}}$. For $n \geq 24 d^4 2^d$, we prove an asymptotic formula for the number of prime solutions to the equation $F(x_1, \ldots, x_n) = 0$, provided $F$ satisfies suitable local conditions.
Let $F \in \mathbb{Z}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d \geq 2$, and let $V_F^*$ denote the singular locus of the affine variety $V(F) = \{ \mathbf{z} \in {\mathbb{A}}^n_{\mathbb{C}}: F(\mathbf{z}) = 0 \}$. %defined by the equation $F (x_1, \ldots, x_n) = 0.$ In this paper, we prove the existence of integer solutions with prime coordinates to the equation $F(x_1, \ldots, x_n) = 0$ provided $F$ satisfies suitable local conditions and $n - \dim V_F^* > d^2 (2d-1) 4^{d} \left( 33 + \frac{37}{2d-1 } \right) + \frac{3 (32 d) (33d) (6d + 1)}{d-1} 2^{d-1} + (12 d + 1) (4 (d-1) 2^d + 1)$. The result is obtained by using the Heath-Brown identity and optimizing various parts of the argument in the author's previous work, which made use of the Vaughan identity and required $n - \dim V_F^* \geq 2^8 3^4 5^2 d^3 (2d-1)^2 4^{d}$.
We use function field analytic number theory to establish the irreducibility and dimension of the moduli space that parameterises morphisms of fixed degree from $\mathbb{P}^2$ to an arbitrary smooth hypersurface of sufficiently small degree.
For any $d \geq 2$, we prove that there exists an integer $n_0(d)$ such that there exists an $n \times n$ magic square of $d^\text{th}$ powers for all $n \geq n_0(d)$. In particular, we establish the existence of an $n \times n$ magic square of squares for all $n \geq 4$, which settles a conjecture of Várilly-Alvarado. All previous approaches had been based on constructive methods and the existence of $n \times n$ magic squares of $d^\text{th}$ powers had only been known for sparse values of $n$. We prove our result by the Hardy-Littlewood circle method, which in this setting essentially reduces the problem to finding a sufficient number of disjoint linearly independent subsets of the columns of the coefficient matrix of the equations defining magic squares. We prove an optimal (up to a constant) lower bound for this quantity.
In this paper, we obtain an asymptotic formula for the number of integral solutions to a system of diagonal equations. We obtain an asymptotic formula for the number of solutions with variables restricted to smooth numbers as well. We improve the required number of variables compared to previous results by incorporating the recent progress on Waring's problem and the resolution of the main conjecture in Vinogradov's mean value theorem.
We study the geometry of the space of rational curves on smooth complete intersections of low degree, which pass through a given set of points on the variety. The argument uses spreading out to a finite field, together with an adaptation to function fields of positive characteristic of work by Rydin Myerson on the circle method. Our work also allows us to handle weak approximation for such varieties.
In this paper, we prove that there does not exist $F \in \mathbb{Q}[x,y]$ of degree $4$ such that $F(\mathbb{Z}^2) = \mathbb{Z}_{\geq 0}$. In particular, this answers a question by John S. Lew and Bjorn Poonen for quartic polynomials.
Let $F \in \mathbb{Z}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d \geq 2$, and $V_F^*$ the singular locus of the hypersurface $\{\mathbf{x} \in \mathbb{A}^n_{\mathbb{C}}: F(\mathbf{x}) = 0 \}$. A longstanding result of Birch states that there is a non-trivial integral solution to the equation $F(\mathbf{x}) = 0$ provided $n > \dim V_F^* + (d-1) 2^d$ and there is a non-singular solution in $\mathbb{R}$ and $\mathbb{Q}_p$ for all primes $p$. In this article, we give a different formulation of this result. More precisely, we replace $\dim V_F^*$ with a quantity $\mathcal{H}_F$ defined in terms of the Hessian matrix of $F$. This quantity satisfies $0 \leq \mathcal{H}_F \leq \dim V_F^*$; therefore, we improve on the aforementioned result of Birch if $\mathcal{H}_F < \dim V_F^*$. We also prove the corresponding result for systems of forms of equal degree.
In this article, we establish an analogue of the dimension growth conjecture, which is regarding the density of rational points on projective varieties, for compact submanifolds of $\mathbb{R}^n$ with non-vanishing curvature. We also establish the convergence theory for the set of simultaneously $ψ$-approximable points lying on a generic hypersurface, thereby settling the generalized Baker-Schmidt problem in the simultaneous setting for generic hypersurfaces. These results are obtained as consequences of an optimal upper bound for the density of rational points near manifolds of the form $\{ (\mathbf{x}, f(\mathbf{x})) \in \mathbb{R}^{d+1}: \mathbf{x} \in B_ν(\mathbf{x}_0) \}$ with non-zero Hessian matrix of $f$ at $\mathbf{x}_0$ and $ν> 0$ sufficiently small.
Let $F \in \mathbb{Z}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d \geq 2$, and let $V_F^*$ denote the singular locus of the affine variety $V(F) = \{ \mathbf{z} \in {\mathbb{C}}^n: F(\mathbf{z}) = 0 \}$. In this paper, we prove the existence of integer solutions with prime coordinates to the equation $F(x_1, \ldots, x_n) = 0$ provided $F$ satisfies suitable local conditions and $n - \dim V_F^* \geq 2^8 3^4 5^2 d^3 (2d-1)^2 4^{d}$. Our result improves on what was known previously due to Cook and Magyar (B. Cook and A. Magyar, `Diophantine equations in the primes'. Invent. Math. 198 (2014), 701-737), which required $n - \dim V_F^*$ to be an exponential tower in $d$.
A semiprime is a natural number which is the product of two (not necessarily distinct) prime numbers. Let $F(x_1, \ldots, x_n)$ be a degree $d$ homogeneous form with integer coefficients. We provide sufficient conditions, similar to those of the seminal work of B. J. Birch, for which the equation $F (x_1, \ldots, x_n) = 0$ has infinitely many integer solutions with semiprime coordinates. Previously it was known, by a result of Á. Magyar and T. Titichetrakun, that under the same hypotheses there exist infinitely many integer solutions to the equation with coordinates that have at most $384 n^{3/2} d (d+1)$ prime factors.
We study the density of rational points on a higher-dimensional orbifold $(\mathbb{P}^{n-1},D)$ when $D$ is a $\mathbb{Q}$-divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy-Littlewood circle method to first study an asymptotic version of Waring's problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov's mean value theorem, due to Bourgain-Demeter-Guth and Wooley.
Let $F \in \mathbb{R}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d > 1$ satisfying $(n - \dim V_{F}^*) > 4$, where $V_F^*$ is the singular locus of $V(F) = \{ \mathbf{z} \in {\mathbb{C}}^n: F(\mathbf{z}) = 0 \}$. Suppose there exists $\mathbf{x}_0 \in (0,1)^n \cap (V(F) \backslash V_F^*)$. Let $\mathbf{t} = (t_1, \ldots, t_n) \in \mathbb{R}^n$. Then for a smooth function $\varpi:\mathbb{R}^n \rightarrow \mathbb{R}$ with its support contained in a small neighbourhood of $\mathbf{x}_0$, we prove $$ \Big{|} \int_{0}^{\infty} \cdots \int_{0}^{\infty} \varpi(\mathbf{x}) x_1^{i t_1} \cdots x_n^{i t_n} e^{2 πi τF(\mathbf{x})} d \mathbf{x} \Big{|} \ll \min \{ 1, |τ|^{-1} \}, $$ where the implicit constant is independent of $τ$ and $\mathbf{t}$.
In this paper we show that a polynomial equation admits infinitely many prime-tuple solutions assuming only that the equation satisfies suitable local conditions and the polynomial is sufficiently non-degenerate algebraically. Our notion of algebraic non-degeneracy is related to the $h$-invariant introduced by W. M. Schmidt. Our results prove a conjecture of B. Cook and Á. Magyar \cite{CM} for hypersurfaces of degrees $2$ and $3$.
Let $\mathbf{f} = (f_1, \ldots, f_R)$ be a system of polynomials with integer coefficients in which the degrees need not all be the same. We provide sufficient conditions for which the system of equations $f_j (x_1, \ldots, x_n) = 0 \ (1 \leq j \leq R)$ satisfies a general local to global type statement, and has a solution where each coordinate is prime. In fact we obtain the asymptotic formula for number of such solutions, counted with a logarithmic weight, under these conditions. We prove the statement via the Hardy-Littlewood circle method. This is a generalization of the work of B. Cook and Á. Magyar, where they obtained the result when the polynomials of $\mathbf{f}$ all have the same degree. Hitherto, results of this type for systems of polynomial equations involving different degrees have been restricted to the diagonal case.
In his paper, W. M. Schmidt obtained an exponential sum estimate for systems of polynomials not including linear polynomials, which was then used to apply the Hardy-Littlewood circle method. We prove an analogous estimate for systems including linear polynomials.
Let $ω$ be a sequence of positive integers. Given a positive integer $n$, we define $$ r_n(ω) = | \{ (a,b)\in \mathbb{N}\times \mathbb{N}\colon a,b \in ω, a+b = n, 0 0$ for all $n$ sufficiently large and, for all $ε> 0$, $$ \lim_{n \rightarrow \infty} \frac{r_n(ω)}{n^ε} = 0. $$ P. Erdős proved this conjecture by showing the existence of a sequence $ω$ of positive integers such that $$ \log n \ll r_n(ω) \ll \log n. $$ In this paper, we prove an analogue of this conjecture in $\mathbb{F}_q[T]$, where $\mathbb{F}_q$ is a finite field of $q$ elements. More precisely, let $ω$ be a sequence in $\mathbb{F}_q[T]$. Given a polynomial $h\in\mathbb{F}_q[T]$, we define $$ r_h(ω) = |\{(f,g) \in \mathbb{F}_q[T]\times \mathbb{F}_q[T] : f,g\in ω, f+g =h, \text{deg } f, \text{deg } g \leq \text{deg } h, f\ne g\}|. $$ We show that there exists a sequence $ω$ of polynomials in $\mathbb{F}_q [T]$ such that $$ \text{deg } h \ll r_h(ω) \ll \text{deg } h $$ for $\text{deg } h$ sufficiently large.
Let $\mathbb{F}_q[t]$ denote the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements. Suppose the characteristic of $\mathbb{F}_q$ is not $2$ or $3$. In this paper, we prove an $\mathbb{F}_q[t]$-analogue of results related to the conjecture of Erdős on the existence of infinite Sidon sequence of positive integers which is an asymptotic basis of order 3. We prove that there exists a $B_2[2]$ sequence of non-zero polynomials in $\mathbb{F}_q[t]$, which is an asymptotic basis of order $3$. We also prove that for any $\varepsilon> 0$, there exists a sequence of non-zero polynomials in $\mathbb{F}_q[t]$, which is a Sidon basis of order $3 + \varepsilon$. In other words, there exists a sequence of non-zero polynomials in $\mathbb{F}_q[t]$ such that any $n \in \mathbb{F}_q[t]$ of sufficiently large degree can be expressed as a sum of four elements of the sequence, where one of them has a degree less than or equal to $\varepsilon \text{deg } n.$