arXiv subjects

Anton Mosunov

Publications and source records attributed to Anton Mosunov.

6 recordsLinked to original sources

arXiv2022

A Lower Bound for the Area of the Fundamental Region of a Binary Form

Let $$ F(x, y) = \prod\limits_{k = 0}^{n - 1}(δ_kx - γ_ky) $$ be a binary form of degree $n \geq 1$, with complex coefficients, written as a product of $n$ linear forms in $\mathbb C[x, y]$. Let $$ h_F = \prod\limits_{k = 0}^{n - 1}\sqrt{|γ_k|^2 + |δ_k|^2} $$ denote the height of $F$ and let $A_F$ denote the area of the fundamental region of $F$: $$ \left\{(x, y) \in \mathbb R^2 \colon |F(x, y)| \leq 1\right\}. $$ We prove that $h_F^{2/n}A_F \geq \left(2^{1 + (r/n)}\right)π$, where $r$ is the number of roots of $F$ on the real projective line $\mathbb R\mathbb P^1$, counting multiplicity.

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arXiv2022

On the Generalization of the Gap Principle

Let $α$ be a real algebraic number of degree $d \geq 3$ and let $β\in \mathbb Q(α)$ be irrational. Let $μ$ be a real number such that $(d/2) + 1 < μ< d$ and let $C_0$ be a positive real number. We prove that there exist positive real numbers $C_1$ and $C_2$, which depend only on $α$, $β$, $μ$ and $C_0$, with the following property. If $x_1/y_1$ and $x_2/y_2$ are rational numbers in lowest terms such that $$ H(x_2, y_2) \geq H(x_1, y_1) \geq C_{1} $$ and $$ \left|α- \frac{x_1}{y_1}\right| < \frac{C_0}{H(x_1, y_1)^μ}, \quad \left|β- \frac{x_2}{y_2}\right| < \frac{C_0}{H(x_2, y_2)^μ}, $$ then either $H(x_2, y_2) > C_{2}^{-1} H(x_1, y_1)^{μ- d/2}$, or there exist integers $s, t, u, v$, with $sv - tu \neq 0$, such that $$ β= \frac{sα+ t}{uα+ v} \quad \text{and} \quad \frac{x_2}{y_2} = \frac{sx_1 + ty_1}{ux_1 + vy_1}, $$ or both. Here $H(x, y) = \max(|x|, |y|)$ is the height of $x/y$. Since $μ- d/2$ exceeds one, our result demonstrates that, unless $α$ and $β$ are connected by means of a linear fractional transformation with integer coefficients, the heights of $x_1/y_1$ and $x_2/y_2$ have to be exponentially far apart from each other. An analogous result is established in the case when $α$ and $β$ are $p$-adic algebraic numbers.

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arXiv2022

Absolute Bound On the Number of Solutions of Certain Diophantine Equations of Thue and Thue-Mahler Type

Let $F \in \mathbb Z[x, y]$ be an irreducible binary form of degree $d \geq 7$ and content one. Let $α$ be a root of $F(x, 1)$ and assume that the field extension $\mathbb Q(α)/\mathbb Q$ is Galois. We prove that, for every sufficiently large prime power $p^k$, the number of solutions to the Diophantine equation of Thue type $$ |F(x, y)| = tp^k $$ in integers $(x, y, t)$ such that $\gcd(x, y) = 1$ and $1 \leq t \leq (p^k)^λ$ does not exceed $24$. Here $λ= λ(d)$ is a certain positive, monotonously increasing function that approaches one as $d$ tends to infinity. We also prove that, for every sufficiently large prime number $p$, the number of solutions to the Diophantine equation of Thue-Mahler type $$ |F(x, y)| = tp^z $$ in integers $(x, y, z, t)$ such that $\gcd(x, y) = 1$, $z \geq 1$ and $1 \leq t \leq (p^z)^{\frac{10d - 61}{20d + 40}}$ does not exceed 1992. Our proofs follow from the combination of two principles of Diophantine approximation, namely the generalized non-Archimedean gap principle and the Thue-Siegel principle.

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arXiv2021

On the Automorphism Group of a Binary Form Associated with Algebraic Trigonometric Quantities

Let $F(x, y)$ be a binary form of degree at least three and non-zero discriminant. In this article we compute the automorphism group $\operatorname{Aut} F$ for four families of binary forms. The first two families that we are interested in are homogenizations of minimal polynomials of $2\cos\left(\frac{2π}{n}\right)$ and $2\sin\left(\frac{2π}{n}\right)$, which we denote by $Ψ_n(x, y)$ and $Π_n(x, y)$, respectively. The remaining two forms that we consider are homogenizations of Chebyshev polynomials of first and second kinds, denoted $T_n(x, y)$ and $U_n(x, y)$, respectively.

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arXiv2021

On the Area of the Fundamental Region of a Binary Form Associated with Algebraic Trigonometric Quantities

Let $F(x, y)$ be a binary form of degree at least three and non-zero discriminant. We estimate the area $A_F$ bounded by the curve $|F(x, y)| = 1$ for four families of binary forms. The first two families that we are interested in are homogenizations of minimal polynomials of $2\cos\left(\frac{2π}{n}\right)$ and $2\sin\left(\frac{2π}{n}\right)$, which we denote by $Ψ_n(x, y)$ and $Π_n(x, y)$, respectively. The remaining two families of binary forms that we consider are homogenizations of Chebyshev polynomials of first and second kinds, denoted $T_n(x, y)$ and $U_n(x, y)$, respectively.

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arXiv2021

On the Area Bounded by the Curve $\prod_{k = 1}^n |x\sin(kπ/n)-y\cos(kπ/n)| = 1$

For a positive integer $n$, let $$ F_n^*(X, Y) = \prod\limits_{k = 1}^n\left(X\sin\left(\frac{kπ}{n}\right) -Y\cos\left(\frac{kπ}{n}\right)\right). $$ In 2000 Bean and Laugesen proved that for every $n \geq 3$ the area bounded by the curve $|F_n^*(x, y)| = 1$ is equal to $4^{1 - 1/n}B\left(\frac{1}{2} - \frac{1}{n}, \frac{1}{2}\right)$, where $B(x, y)$ is the beta function. We provide an elementary proof of this fact based on the polar formula for the area calculation. We also prove that $$ F_n^*(X, Y) = 2^{1 - n}\sum\limits_{\substack{1 \leq k \leq n\\\text{$k$ is odd}}}(-1)^{\frac{k - 1}{2}}\binom{n}{k}X^{n - k}Y^k $$ and demonstrate that $\ell_n = 2^{n - 1 - ν_2(n)}$ is the smallest positive integer such that the binary form $\ell_n F_n^*(X, Y)$ has integer coefficients. Here $ν_2(n)$ denotes the $2$-adic order of $n$.

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