Real roots of random polynomials [Link]
Description: Random polynomials arise when the coefficients of a polynomial are treated as random variables. This seemingly simple change transforms a fundamental mathematical object into a powerful tool connecting mathematics, physics, and other fields. Since the coefficients are random, the number of real roots of these random polynomials itself becomes a random variable. A central challenge in this area concerns the behavior of this random variable: How many real roots are there typically? How are these roots distributed along the real number line? And how do their positions relate to each other? This project investigates these questions for a specific type of random polynomial: generalized Kac polynomials. We will explore the correlations between the real roots of these polynomials by combining theoretical insights from analysis and probability with hands-on computer simulations. The work is designed to be accessible, even for participants with no prior programming experience.
Dates: May 12 - June 27, 2025
Undergraduates: Farid Ahmadov, Ayush Khadka, Melanie Fouque, and Connor Shanley.
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