arXiv subjects

Roy Shmueli

Publications and source records attributed to Roy Shmueli.

4 recordsLinked to original sources

arXiv2024

Irreducibility of Polynomials with Square Coefficients over Finite Fields

We study a random polynomial of degree $n$ over the finite field $\mathbb{F}_q$, where the coefficients are independent and identically distributed and uniformly chosen from the squares in $\mathbb{F}_q$. Our main result demonstrates that the likelihood of such a polynomial being irreducible approaches $1/n + O(q^{-1/2})$ as the field size $q$ grows infinitely large. The analysis we employ also applies to polynomials with coefficients selected from other specific sets.

math.NT↗

arXiv2022

The probability that a p-adic random étale algebra is an unramified field

We study the random étale algebra generated by a random polynomial with i.i.d. coefficients distributed according to Haar measure normalized on $\mathbb{Z}_p$. We determine the probability that this random algebra is an unramified field, explicitly. In addition, we prove a private case of a conjecture made by Bhargava, Cremona, Fisher, and Gajović. More precisely, we show that this probability is a rational function of $p$ that is invariant under replacing $p$ by $1/p$.

math.NT↗

arXiv2022

The Number of Roots of a Random Polynomial over The Field of $p$-adic Numbers

We study the roots of a random polynomial over the field of p-adic numbers. For a random monic polynomial with coefficients in $\mathbb{Z}_p$, we obtain an asymptotic formula for the factorial moments of the number of roots of this polynomial. In addition, we show the probability that a random polynomial of degree $n$ has more than $\log n$ roots is $O\big(n^{-K}\big)$ for some $K > 0$.

math.NT↗

arXiv2021

The Expected Number of Roots over The Field of p-adic Numbers

We study the roots of a random polynomial over the field of $p$-adic numbers. For a random monic polynomial with i.i.d. coefficients in $\mathbb{Z}_p$, we obtain an estimate for the expected number of roots of this polynomial. In particular, if the coefficients take the values $\pm1$ with equal probability, the expected number of $p$-adic roots converges to $\left(p-1\right)/\left(p+1\right)$ as the degree of the polynomial tends to $\infty$.

math.NT↗