arXiv subjects
Publications and source records attributed to Sajad A. Sheikh.
In this paper, we investigate the algebraic counterpart of the Fundamental Theorem of Algebra. We explore the concept of real closed fields and quadratic forms. We show, by means of Galois theory, that $F(\sqrt{-1})$ is algebraically closed if $F$ is real-closed. Lastly, we explain the algebraic closure of $\mathbb R(\sqrt{-1})=\mathbb C$ by demonstrating the real-closeness of $\mathbb R$.
Classical zero-localization theorems give deterministic certificates that all zeros of a polynomial lie in a prescribed disk, annulus, or related region. When the coefficients are random, each such deterministic certificate becomes a random variable on coefficient space. This paper develops a finite-degree certificate method for random polynomial localization and extends the author's earlier joint work with Mir \cite{SheikhMir2024}. The main result concerns Gaussian polynomials with random leading coefficient: the Cauchy ratios are marginally standard Cauchy, yet they are dependent through their common denominator. We derive the exact dependence-aware certificate integral and prove that its inverse confidence radius has order \(\sqrt{\log n}\), while a fictitious independent-Cauchy model has order \(n\). We also obtain monic coefficient-law certificates, sub-Weibull confidence radii, annular certificates via reversal, Rouché--Chernoff certificates, and an optimized scaled Cauchy envelope. A reproducible Monte Carlo study for monic Gaussian polynomials compares the classical Cauchy radius, the optimized Cauchy envelope, the annular certificate, and the Rouché radius. Across \(5000\) samples for each of \(n=20,50,100\), the Rouché certificate gives the sharpest outer radii, the optimized Cauchy envelope substantially improves the classical Cauchy radius, and all tested certificates satisfy their deterministic containment inequalities numerically.
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 article presents some interesting and novel results concerning the average modulus of random polynomials on the unit circle and the unit disc, with coefficients distributed as standard normal variates. The paper also introduces new results concerning the bounds of the maximum modulus of random polynomials with coefficients distributed as independently as Gaussian and uniform variates, utilizing probability principles to derive findings about the likelihood of the maximum modulus exceeding a specific threshold, using Markov inequality as the primary probabilistic tool. These findings and the approach can potentially initiate the study of a rich class of problems concerning the norms of random polynomials.