This paper presents a comprehensive exploration of a probabilistic adaptation of the Eneström–Kakeya theorem, applied to random polynomials featuring various coefficient distributions. Unlike the deterministic rendition of the theorem, our study dispenses with the necessity of any specific coefficient order. Instead, we consider coefficients drawn from a spectrum of sets with diverse probability distributions, encompassing finite, countable, and uncountable sets. Furthermore, we provide a result concerning the probability of failure of Schur stability for a random polynomial with coefficients distributed independently and identically as standard normal variates. We also provide simulations to corroborate our results.

This paper presents a significant extension of perturbation theory techniques for estimating the roots of polynomials. Building upon foundational results and recent work by Pakdemirli and Yurtsever, as well as taking inspiration from the concept of probabilistic bounds introduced by Sheikh et al., we develop and prove several novel theorems that address a wide range of polynomial structures. These include polynomials with multiple large coefficients, coefficients of different orders, alternating coefficient orders, large linear and constant terms, and exponentially decreasing coefficients. Among the key contributions is a theorem that establishes an upper bound on the expected maximum modulus of the zeros of polynomials with uniformly distributed perturbations in their coefficients. The theorem considers the case where all but the leading coefficient receive a uniformly and independently distributed perturbation in the interval $[-1,1]$. Our approach provides a comprehensive framework for estimating the order of magnitude of polynomial roots based on the structure and magnitude of their coefficients without the need for explicit root-finding algorithms. The results offer valuable insights into the relationship between coefficient patterns and root behavior, extending the applicability of perturbation-based root estimation to a broader class of polynomials. This work has potential applications in various fields, including random polynomials, control systems design, signal processing, and numerical analysis, where quick and reliable estimation of polynomial roots is crucial. Our findings contribute to the theoretical understanding of polynomial properties and provide practical tools for engineers and scientists dealing with polynomial equations in diverse contexts.

This paper introduces the notion of probabilistic zero bounds for random polynomials. It presents new results regarding the probabilistic bounds of random polynomials whose coefficients are independently and identically distributed as standard normal variates. Additionally, the paper provides a clear exposition of the developed methodology. To establish our results, we develop a novel approach utilizing the classical Cauchy’s bounds for the zeros of a deterministic polynomial with complex coefficients. We also corroborate our analytical results with extensive simulations. The methodology developed in the paper can potentially be applied to a broad class of problems regarding bounds and the distribution of zeros in the theory of random polynomials.

This paper investigates various aspects of the distribution of roots and critical points of a complex polynomial, including their variance and the relationships between their moduli using an inequality due to de Bruijn. Making use of two other inequalities-again due to de Bruijn-we derive two probabilistic results concerning upper bounds for the average moduli of the imaginary parts of zeros and those of critical points, assuming uniform distribution of the zeros over a unit disc and employing the Markov inequality. The paper also provides an explicit formula for the variance of the roots of a complex polynomial for the case when all the zeros are real. In addition, for polynomials with uniform distribution of roots over the unit disc, the expected variance of the zeros is computed. Furthermore, a bound on the variance of the critical points in terms of the variance of the zeros of a general polynomial is derived, whereby it is established that the variance of the critical points of a polynomial cannot exceed the variance of its roots. Finally, we conjecture a relation between the real parts of the zeros and the critical points of a polynomial.

We develop a terminal-explicit geometric theory for the algebraic zero sets associated with the Riemann–Liouville and Caputo fractional derivatives of a complex polynomial. Expanding the polynomial about an arbitrary complex terminal a reduces both operators to gamma-weighted polynomial transforms. The coordinate shift $w=z-a$ is algebraically equivalent to a zero-terminal formulation; the terminal dependence re-enters through the shifted coefficients and through the pullback of the resulting geometry to the original z-plane. We derive exact barycenter identities, terminal-centered disk bounds, multiset convergence at the endpoint orders, first-order deformation formulas for simple roots, and solvable examples. The disk bounds provide a star-shaped localization framework rather than a convex-hull theorem. A regular-polygon family yields an exact homothety for the Riemann–Liouville roots, while numerical examples show curved generic trajectories and a nondegenerate Caputo evolution. The terminal therefore acts as a distinguished geometric anchor, with radial attraction occurring under additional algebraic symmetry.