Min Sha

The arithmetic of Carmichael quotients

Carmichael quotients for an integer

Functional graphs of polynomials over finite fields

m\ge 2

On functional graphs of quadratic polynomials

m≥2 are introduced analogous to Fermat quotients, by using Carmichael function

Bounding the j-invariant of integral points on certain modular curves

\lambda (m)

Correction to: The arithmetic of Carmichael quotients

λ(m). Various properties of these new quotients are investigated, such as basic arithmetic properties, sequences derived from Carmichael quotients, Carmichael-Wieferich numbers, and so on. Finally, we link Carmichael quotients to perfect nonlinear functions.

The Sato–Tate distribution in thin parametric families of elliptic curves

In this paper, we concentrate on counting and testing dominant polynomials with integer coefficients. A polynomial is called dominant if it has a simple root whose modulus is strictly greater than the moduli of its remaining roots. In particular, our results imply that the probability that what is known as the dominant root assumption holds for a random monic polynomial with integer coefficients tends to 1 in some setting. However, for arbitrary integer polynomials it does not tend to 1. For instance, the proportion of dominant quadratic integer polynomials of height H among all quadratic integer polynomials tends to (41 + 6log 2)/72 as H → ∞. Finally, we design some algorithms to test whether a given polynomial with integer coefficients is dominant without finding the polynomial’s roots.