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Idrees Qasim

Publications and source records attributed to Idrees Qasim.

7 recordsLinked to original sources
arXiv2026

On Turan Type inequality for Quaternionic Canonical Generalized Polynomials

This paper establishes Tur\'an-type inequalities for quaternionic canonical generalized polynomials with all zeros in the closed unit ball of radius $k \geq 1$. We extend the classical inequality due to Govil from the complex setting to the quaternionic framework. We prove that for certain classes of polynomials, the inequality \[\|P'\| \geq \frac{n}{1+k^n}\|P\|\] holds for all $k \geq 1$, where $n$ is the degree of the polynomial. Our main results provide sharp derivative estimates for quaternionic canonical generalized polynomials under specific algebraic conditions on their zeros. These findings contribute to the ongoing research of extending classical polynomial inequalities to the noncommutative setting of quaternions.

math.CV↗
arXiv2026

On the Right Eigenvalues of the Quaternionic Matrix Polynomials

This paper establishes new upper bounds for the right eigenvalues of monic matrix polynomials over the quaternion division algebra. The noncommutative nature of quaternion multiplication presents fundamental challenges in eigenvalue analysis, distinguishing this problem from the classical complex case. We use spectral norm inequalities for partitioned quaternionic matrices and apply them to quaternionic block matrices associated with monic matrix polynomials. By analyzing the structure of powers of these companion matrices we derive progressively sharper bounds for the right eigenvalues. Consequently, these bounds give bounds for the zeros of quaternionic polynomials.

math.CV↗
arXiv2026

Bounds for the Zeros of Polynomials over Quaternion Division Algebra

Locating the zeros of quaternionic polynomials is a fundamental problem with significant implications across scientific and engineering disciplines, yet the noncommutative nature of quaternion multiplication makes it fundamentally more complex than the classical complex case. In this paper, we develop new bounds for the zeros of polynomials with quaternionic coefficients. We establish spectral norm inequalities for quaternionic matrices, particularly those of a partitioned form. These inequalities are applied to specialized quaternionic companion matrices to derive novel upper bounds for the zeros of the original polynomial. By establishing novel spectral norm inequalities for partitioned quaternionic matrices and utilizing the structural properties of companion matrices and their higher powers, we derive unexplored upper bounds for the zeros of quaternionic polynomials. Our bounds are systematically sharper than existing results and provide a unified framework for zero localization in the quaternionic setting.

math.CV↗
arXiv2026

Bounds for the Zeros of Quaternionic Polynomials via Matrix Methods

In this paper, we derive new bounds for the zeros of quaternionic polynomials by applying localization theorems, which includes Gershgorin-type theorems for the left eigenvalues of matrices of left monic quaternionic polynomials. These results yield sharper estimates compared to existing bounds, including improvements upon Cauchy, Fujiwara and Opfer's classical bounds. Second, we develop a matrix norm approach utilizing block matrix techniques and spectral norm estimates for a specially constructed auxiliary poly nomial. This method provides additional upper bounds for polynomial zeros through careful analysis of the companion matrix's spectral radius. The comparison between the new bounds and some existing bounds have been illustrated with several examples. At the end of the paper we have given an algorithm. We have also given a Python code that predicts, for a given input which theorem will yield the sharpest upper bound. The combination of these approaches enhances the theoretical toolkit for analyzing quaternionic polynomials and offers potential applications in numerical methods, signal processing, and quaternionic quantum mechanics where zero location problems naturally arise.

math.CV↗
arXiv2024

Bounds for the zeros of Bicomplex Polynomials using matrix method

In this paper we investigate bounds for the zeros of a bicomplex polynomial using matrix method. In particular, we find analogue of Gershgorin disk theorem, Cauchy Theorem, theorem of Fujiwara, Walsh and other theorems concerning to zeros of a polynomial to bicomplex polynomials.

math.CV↗