arXiv subjects

Or Ben-Porath

Publications and source records attributed to Or Ben-Porath.

3 recordsLinked to original sources

arXiv2026

Counting Polynomials via Galois Actions on Root Subsets

This paper studies the number of monic integer polynomials $f$ of height at most $H$ whose Galois group, endowed with the action on the roots, is isomorphic to a prescribed permutation group $(G,Ω)$. New upper bounds are obtained for several families of groups: transitive subgroups of the wreath product $S_m\wr S_r$ in the primitive action; $k$-homogeneous subgroups of $S_m$ in the action on $k$-subsets of $\{1,\ldots,m\}$; $k$-transitive subgroups of $S_m$ in the action on $k$-tuples of distinct elements of $\{1,\ldots,m\}$. Almost all finite groups in their regular permutation representation are also treated.

math.NT↗

arXiv2024

Probabilistic Galois Theory -- The Square Discriminant Case

The paper studies the probability for a Galois group of a random polynomial to be $A_n$. We focus on the so-called large box model, where we choose the coefficients of the polynomial independently and uniformly from $\{-L,\ldots, L\}$. The state-of-the-art upper bound is $O(L^{-1})$, due to Bhargava. We conjecture a much stronger upper bound $L^{-n/2 +ε}$, and that this bound is essentially sharp. We prove strong lower bounds both on this probability and on the related probability of the discriminant being a square.

math.NT↗

arXiv2020

On the Galois Theory of Generalized Laguerre Polynomials and Trimmed Exponential

Inspired by the work of Schur on the Taylor series of the exponential and Laguerre polynomials, we study the Galois theory of trimmed exponentials $f_{n,n+k}=\sum_{i=0}^{k} \frac{x^{i}}{(n+i)!}$ and of the generalized Laguerre polynomials $L^{(n)}_k$ of degree $k$. We show that if $n$ is chosen uniformly from $\{1,\ldots, x\}$, then, asymptotically almost surely, for all $k\leq x^{o(1)}$ the Galois groups of $f_{n,n+k}$ and of $L_{k}^{(n)}$ are the full symmetric group $S_k$.

math.NT↗