Functions and Analysis

2602 Submissions

[2] viXra:2602.0053 [pdf] submitted on 2026-02-06 21:54:21

Unified Proof of Sendov’s Conjecture via the Potential Stability Paradigm: Topological Rigidity and Field Monotonicity

Authors: Francisco Petitti
Comments: 43 Pages.

Sendov’s Conjecture (1962) asserts that if all zeros {z1, ..., zn} of a polynomial P(z) of degree n >= 2 lie within the closed unit disk D, then each zero zk is at a distance no greater than 1 from at least one critical point of P(z). While recent advancements have confirmed the conjecture for sufficiently large n, a unified proof for intermediate degrees has remained an open challenge due to the complex discrete interactions of root configurations.This paper provides a complete resolution of Sendov’s Conjecture for all n >= 2 by reformulating the problem within the framework of Logarithmic Potential Dynamics. We treat the critical points as equilibrium positions in a complex force field generated by the zeros of the polynomial. By isolating a fixed root z1 = a in [0, 1], the total force F(z) = P'(z)/P(z) is decomposed into a Local Attraction Force and a Collective Cloud Repulsion Force.By employing a Laurent series expansion of the cloud force, we derive a universal upper bound for the remainder Rn(z) based on the majorization of the spectral moments (power sums) of the roots constrained within D. We prove that the repulsion magnitude is strictly dominated by the geometric decay of the local force at the boundary |z - a| = 1.0. Furthermore, by analyzing the Hessian of the potential, we establish the strict monotonicity of the radial force field in the region r >= 1.0, thereby precluding the existence of equilibrium points beyond the unit radius of any given zero. This analytical framework bridges the gap between asymptotic analysis and discrete root geometry, confirming the conjecture for the general case.
Category: Functions and Analysis

[1] viXra:2602.0032 [pdf] submitted on 2026-02-05 20:19:16

Expansion of the Number π Into a Series of Trigonometric Functions and Its Connection with the Riemann Zeta Function

Authors: Bohdan Dykyi
Comments: 8 Pages. (Note by viXra Admin: Please cite and list scientific references)

In this work, series expansions for negative and positive integer powers of the number π are derived using Viète’s infinite product and differentiation techniques. A representation of these powers in terms of trigonometric series involving tangent functions is obtained. Furthermore, a connection between these expansions and the values of the Riemann zeta function at even arguments is established. Explicit formulas for the reciprocal values of the zeta function are presented. Several illustrative examples are provided.
Category: Functions and Analysis