arXiv subjects
Publications and source records attributed to Dzianis Kaliada.
In the paper we study the distribution of the discriminant $D(P)$ of polynomials $P$ from the class $\mathcal{P}_{n}(Q)$ of all integer polynomials of degree $n$ and height at most $Q$. We evaluate the asymptotic number of polynomials $P\in \mathcal{P}_{n}(Q)$ having all the roots real and satisfying the inequality $|D(P)|\le X$ as $Q\to\infty$ and $X/Q^{2n-2}\to 0$.
In the paper, we study the asymptotic distribution of real algebraic integers of fixed degree as their na\"ıve height tends to infinity. For an arbitrary interval $I \subset \mathbb{R}$ and sufficiently large $Q>0$, we obtain an asymptotic formula for the number of algebraic integers $α\in I$ of fixed degree $n$ and na\"ıve height $H(α)\le Q$. In particular, we show that the real algebraic integers of degree $n$, with their height growing, tend to be distributed like the real algebraic numbers of degree $n-1$. However, we reveal two symmetric "plateaux", where the distribution of real algebraic integers statistically resembles the rational integers.
For $B\subset\mathbb{R}^k$ denote by $Φ_k(Q;B)$ the number of ordered $k$-tuples in $B$ of real conjugate algebraic numbers of degree $\leq n$ and naive height $\leq Q$. We show that $$ Φ_k(Q;B) = \frac{(2Q)^{n+1}}{2ζ(n+1)} \int_{B} ρ_k(\mathbf{x})\,d\mathbf{x} + O\left(Q^n\right),\quad Q\to \infty, $$ where the function $ρ_k$ will be given explicitly. If $n=2$, then an additional factor $\log Q$ appears in the reminder term.
In this paper we study the distribution of the real algebraic numbers. Given an interval $I$, a positive integer $n$ and $Q>1$, define the counting function $Φ_n(Q;I)$ to be the number of algebraic numbers in $I$ of degree $n$ and height $\le Q$. Let $I_x = (-\infty,x]$. The distribution function is defined to be the limit (as $Q\to\infty$) of $Φ_n(Q;I_x)$ divided by the total number of real algebraic numbers of degree $n$ and height $\le Q$. We prove that the distribution function exists and is continuously differentiable. We also give an explicit formula for its derivative (to be referred to as the distribution density) and establish an asymptotic formula for $Φ_n(Q;I)$ with upper and lower estimates for the error term in the asymptotic. These estimates are shown to be exact for $n \ge 3$. One consequence of the main theorem is the fact that the distribution of real algebraic numbers of degree $n \ge 2$ is non-uniform.
In the paper, we study the asymptotic distribution of real algebraic integers of fixed degree as their naive height tends to infinity. Let $I \subset \mathbb{R}$ be an arbitrary bounded interval, and $Q$ be a sufficiently large number. We obtain an asymptotic formula for the count of algebraic integers $α$ of fixed degree $n$ and naive height $H(α)\le Q$ lying in $I$. In this formula, we estimate the order of the error term from above and below. We show that algebraic integers of degree $n$ are distributed asymptotically like algebraic numbers of degree $(n-1)$ as the upper bound $Q$ of heights tends to infinity.
For a region $Ω\subset\mathbb{C}$ denote by $Ψ(Q;Ω)$ the number of complex algebraic numbers in $Ω$ of degree $\leq n$ and naive height $\leq Q$. We show that $$ Ψ(Q;Ω)=\frac{Q^{n+1}}{2ζ(n+1)}\int_Ωψ(z)\,ν(dz)+O\left(Q^n \right),\quad Q\to\infty, $$ where $ν$ is the Lebesgue measure on the complex plane and the function $ψ$ will be given explicitly.
We give an explicit formula for the correlation functions of real zeros of a random polynomial with arbitrary independent continuously distributed coefficients.