Alexei Entin

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math.NT2025

Lower Bounds on the Least Common Multiple of a Polynomial Sequence and its Radical

Alexei Entin

Cilleruelo conjectured that for an irreducible polynomial $f \in Z [X]$ of degree $d \geq 2$ , denoting $L_{f} (N) = lcm (f (1), f (2), \dots f (N))$ one has $$\log L_f(n)\…

math.NT2025

The Least Common Multiple of Polynomial Values over Function Fields

Alexei Entin, Sean Landsberg

Cilleruelo conjectured that for an irreducible polynomial $f \in Z [X]$ of degree $d \geq 2$ one has $$\log\left[\mathrm{lcm}(f(1),f(2),\ldots f(N))\right]\sim(d-1)N\log N$…

math.NT2024

Galois groups of random polynomials over the rational function field

Alexei Entin

For a fixed prime power $q$ and natural number $d$ we consider a random polynomial $f = x^{n} + a_{n - 1} (t) x^{n - 1} + \dots + a_{1} (t) x + a_{0} (t) \in F_{q} [t] [x]$ with $a_{i}$ drawn uniformly…

math.NT2024

Automorphism Groups of Finite Extensions of Fields and the Minimal Ramification Problem

Alexei Entin

We study the following question: given a global field $F$ and finite group $G$ , what is the minimal $r$ such that there exists a finite extension $K / F$ with $\mathrm{Aut}(K/F)\cong…

math.NT2024

Probabilistic Galois Theory in Function Fields

Alexei Entin, Alexander Popov

We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial $f = y^{n} + \sum_{i = 0}^{n - 1} a_{i} (x) y^{i} \in F_{q} [x] [y]$ wit…