The $p$-adic limits of iterated $p$-power cyclic resultants of multivariable polynomials
Hyuga Yoshizaki
TL;DR
The paper proves the $p$-adic convergence of the iterated $p$-power cyclic resultants for multivariable polynomials, extending known one-variable results to the multivariate setting via a multivariable resultant framework $r_{n_1,\dots,n_d}(f)$. It develops a theory of $d$-tuple sequences to establish convergence in $\mathbb{Z}_p$ and convergence of the non-$p$-parts, with clear criteria based on $f(1,\dots,1)$. As an application, it proves the $p$-adic convergence of the non-$p$-parts of the sizes of the first homology groups in branched $\mathbb{Z}_p^d$-coverings of links, and identifies the limit with the relative $p$-adic torsion in a specific case, following Kionke’s invariant. Explicit $p$-adic limits are computed for twisted Whitehead links, illustrating the deep link between multivariable Alexander theory, $p$-adic analysis, and topological invariants.
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.