The $p$-adic limits of iterated $p$-power cyclic resultants of multivariable polynomials

Abstract

Let $p$ be a prime number. The $p$-power cyclic resultant of a polynomial is the determinant of the Sylvester matrix of $t^{p^n}-1$ and the polynomial. It is known that the sequence of $p$-power cyclic resultants and its non-$p$-parts converge in ${\mathbb{Z}}_p$. This article shows the $p$-adic convergence of the iterated $p$-power cyclic resultants of multivariable polynomials. As an application, we show the $p$-adic convergence of the torsion numbers of ${\mathbb{Z}}_p^d$-coverings of links. We also explicitly calculate the $p$-adic limits for the twisted Whitehead links as concrete examples. Moreover, in a specific case, we show that our $p$-adic limit of torsion numbers coincides with the $p$-adic torsion, which is a homotopy invariant of a CW-complex introduced by S. Kionke.

18 pages, 1 figure