Publications

2022

1. Preprint

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

   Shmueli, Roy

   arXiv preprint arXiv:2211.12995 , November 2022

   Abs arXiv PDF

   We study the random étale algebra generated by a random polynomial with i.i.d. coefficients distributed according to Haar measure normalized on 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.

2. Preprint

   The number of roots of a random polynomial over the field of p-adic numbers

   Shmueli, Roy

   arXiv preprint arXiv:2204.03473 , April 2022

   Abs arXiv PDF

   We study the roots of a random polynomial over the field of p-adic numbers. For a random monic polynomial with coefficients in 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(n-K) for some K>0.

2021

1. Journal

   The expected number of roots over the field of p-adic numbers

   Shmueli, Roy

   International Mathematics Research Notices , November 2021

   Abs HTML PDF

   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 Zp, we obtain an estimate for the expected number of roots of this polynomial. In particular, if the coefficients take the values ±1 with equal probability, the expected number of p-adic roots converges to (p-1)/(p+1) as the degree of the polynomial tends to ∞.