Quadratic equations over p-adic fields and their applications in statistical mechanics

Abstract

(0) or not. This question was open even for a quadratic equation. In this paper, by using the Newton polygon, we provide solvability criteria for quadratic equations over the domains mentioned above for all odd primes p. We also study the number of roots of quadratic equations over all domains given above. This study allows us to present a local description of roots of quadratic equations over p-adic fields whenever p > 2.

The study on general cubic equations over p-adic fields

“…The solvability criterion for quadratic equations over the p-adic field was provided in all classical p-adic analysis books. Moreover, a local description of roots of the quadratic equation was also studied in the paper [29]. Recently, in the series of papers [20][21][22], [26-32, 34, 36], the solvability criteria and the number of roots of depressed cubic equations over the p-adic field were studied.…”

“…In fact, all results in this paper are extension and unification of the previous results. Meanwhile, applications of quadratic and cubic equations in the p-adic lattice models of statistical mechanics were presented in the papers [1,2,23,25,29,30,33]. We would like to stress that quadratic and cubic equations have naturally arisen in the investigations of p-adic Gibbs measures of Potts models on Cayley trees.…”

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

“…This paper is a continuation of previous studies and we are aiming to locally describe all roots of cubic equations over the p−adic field for p > 3. Applications of quadratic and cubic equations in the p−adic lattice models of statistical mechanics were presented in the papers [11]- [13], [18], [24]. The solvability criterion and the number of roots of bi-quadratic equations are also studied in the papers [22,23].…”

The Dynamics of The Potts-Bethe Mapping over $\mathbb{Q}_p:$ The Case $p\equiv 2 \ (mod \ 3) $

“…The existence of p−adic Gibbs measures as well as the phase transition for some p−adic lattice models were established in [27,28,29,35,36,41]. Recently [42,47,48], all translation-invariant p−adic Gibbs measures of the p−adic Potts model on the Cayley tree of order two and three were described by studying allocation of roots of quadratic and cubic equations over some domains of the p−adic fields. However, in the p−adic case, due to lack of the convex structure of the set of p−adic (quasi) Gibbs measures, it was quite difficult to constitute a phase transition with some features of the set of p−adic (quasi) Gibbs measures.…”

“…To the best of our knowledge, the little attention was given to this problem in the literature. Recently, this problem was studied for monomial equations [34], quadratic equations [47], depressed cubic equations for primes p > 3 in [30,31,50] and for primes p = 2, 3 in [44,45,46] and bi-quadratic equations [49].…”