This article is a natural continuation of our previous works [Dutta 2021] and 2022]. In this article, we employ similar ideas as in [Alexandru, Popescu Zaharescu 1990] to provide an estimate IC(K(X)∣K,v) when (K(X)∣K,v) valuation algebraic extension. Our central result analogue 2022, Theorem 1.3]. We further construction complete sequence key polynomials for v over K the setting extensions.

A recurrent sequence of polynomials is a sequence of polynomials that satisfies a linear recursion with fixed polynomial coefficients. Our paper proves that the sequence of Newton polytopes of a recurrent sequence of polynomials is quasi-linear. Our proof uses the Lech-Mahler-Skolem theorem of p-adic analytic number theory with recent results in tropical geometry. A subsequent paper lists some ...

We study the asymptotic distribution of zeros for the random polynomials Pn(z) = ∑n k=0 AkBk(z), where {Ak}k=0 are non-trivial i.i.d. complex random variables. Polynomials {Bk}k=0 are deterministic, and are selected from a standard basis such as Szegő, Bergman, or Faber polynomials associated with a Jordan domain G bounded by an analytic curve. We show that the zero counting measures of Pn conv...

—Many key agreement protocols based on chaotic maps have been proposed to guarantee secure communications in the network. Recently, Wang et al. proposed a new key agreement protocol based on chaotic maps. But some issues exist in this protocol, such as anonymity, key distribution and management, and password problems. In order to improve the performance of Wang et al.’s protocol, this paper pr...

In this paper, we solve a certain family of diophantine equations associated with a family of cyclic quartic number fields. In fact, we prove that for n ≤ 5× 106 and n ≥ N = 1.191× 1019, with n, n+ 2, n2 + 4 square-free, the Thue equation Φn(x, y) = x 4 − nxy − (n + 2n + 4n+ 2)xy − nxy + y = 1 has no integral solution except the trivial ones: (1, 0), (−1, 0), (0, 1), (0,−1).

in this paper we establish several polynomials similar to bernstein's polynomials and several refinements of  hermite-hadamard inequality for convex functions.