On Polynomials with Only Real Roots

Mathematical Research | 2025–2026

A research project investigating structural properties of polynomials whose roots are all real.

What I Explored

I studied properties of real-rooted polynomials and higher-order derivatives, developing and verifying 15+ lemmas and propositions and reviewing 25+ academic sources.

The work also explored connections between polynomial root behavior and stability analysis, particularly concepts relevant to characteristic equations and control systems.

My Contribution

  • Developed and verified mathematical lemmas and propositions
  • Reviewed academic literature and related results
  • Contributed to theoretical extensions and generalizations
  • Worked on proof formalization and manuscript development
  • Explored connections between polynomial theory and engineering stability

Research Reflection

This project changed the way I think about mathematics. A mathematical object that initially seemed purely theoretical can become a useful language for describing and understanding real-world systems.

Elastic Wave Dynamics and Stability in Mechanical Systems

Student Research | 2026–Present

An investigation into wave propagation and dynamic behavior in a mechanical system involving a spherical body suspended on a rubber band and undergoing torsional oscillations.

The Question

How can mathematical relationships between physical characteristics predict the behavior of a dynamic mechanical system?

I developed mathematical models connecting variables such as weight, radius, frequency, and oscillatory behavior, and investigated the formation of standing waves within the system.

Research Process

Model → Experiment → Measurement → Comparison → Refinement

Theoretical predictions were compared against experimental observations, with differences of less than 2 ms between predicted and observed standing-wave formation times.

What I Am Learning

This research has allowed me to explore the relationship between differential equations, mechanical dynamics, experimental measurement, and system stability — ideas that connect naturally to my interest in robotics and engineering.

Mathematical Research & Publications

My research experience has also included working with academic literature, mathematical proofs, and research manuscripts.

Research interests include:

  • Real-rooted polynomials
  • Stability analysis
  • Mathematical modeling
  • Differential equations
  • Mechanical dynamics
  • Wave propagation
  • Control systems

I am interested in research not only as a way to find answers, but as a way to learn how to ask better questions.