Local Descriptions of Roots of Cubic Equations over P-adic Fields

Abstract

The most frequently asked question in the p−adic lattice models of statistical mechanics is that whether a root of a polynomial equation belongs to domains Z

The study on general cubic equations over p-adic fields

“…There are two ways to handle this problem. One way to do it is that we first deduce the general cubic equation to the depressed one, then we apply the results of the papers [20] and [34]. However, this way does not work for domains Z * p , Z p \ Z * p , Q p \ Z p in general.…”

“…We have to care on some special study in the general case. Thus, the main results of this paper cannot be derived from the papers [20,34] (a detailed explanation is given in the next section). In fact, all results in this paper are extension and unification of the previous results.…”

“…This paper is a continuation of the previous studies and we are aiming to study roots of a general cubic equation over the p-adic field for p > 3. It is worth mentioning that any cubic equation can be deduced to a depressed one by suitable linear transformation and a local description of roots of a depressed cubic equation over domains Z * p , Z p \ Z * p , Q p \ Z p , Q p has been already studied in [20,34]. However, by means of results of the papers [20,34], we cannot still derive a local description of roots of a general cubic equation over domains…”

Translation‐invariant generalized P‐adic Gibbs measures for the Ising model on Cayley trees

“…The completion of Q with respect to the p-adic norm defines the p-adic field Q p . 58 An arbitrary p-adic number x ≠ 0 has a unique representation (which is called its canonical form):…”

“…We point out that finding fixed points in the p-adic setting is not an easy job compared to the real case (see previous studies 50,[54][55][56][57]71 for the differences). It is well-known 58 that, in the real case, the existence of a phase transition implies the singularity of limiting Gibbs measures. However, in the p-adic situation, we are able to demonstrate that the generalized p-adic Gibbs measures do not exhibit the mentioned type of singularity (such kind of phenomena is called a strong phase transition).…”