Algebraic characterization of polynomials whose zeros lie in certain algebraic domains

Area of Science:

• Complex analysis
• Algebraic geometry
• Numerical analysis

Background:

• Understanding the location of polynomial zeros is crucial in various scientific fields.
• Existing criteria for zero location are often specific to certain regions or computationally intensive.
• Classical results by Hermite, Hurwitz, and Lyapunov provide foundational methods.

Purpose of the Study:

• To introduce a new, universally applicable algebraic criterion for polynomial zero location.
• To demonstrate the criterion's effectiveness for regions such as circles and half-planes.
• To show that the new criterion encompasses and generalizes existing classical results.

Main Methods:

• Development of a novel algebraic criterion based on polynomial coefficients.
• Application of the criterion to specific algebraic regions (circles, half-planes).
• Demonstration of the criterion's ability to recover known results as special cases.

Main Results:

• A new, effectively computable algebraic criterion for determining if all zeros of a polynomial lie within a specified region T.
• The criterion is shown to be applicable to important regions like circles and half-planes.
• The criterion unifies and generalizes classical results from Hermite, Hurwitz, Lyapunov, and Schur-Cohn.

Conclusions:

• The new algebraic criterion offers a powerful and unified approach to analyzing polynomial zero locations.
• Its computational effectiveness and broad applicability make it a valuable tool in complex analysis and stability theory.
• This work extends the theoretical framework for understanding polynomial behavior in the complex plane.