Expanding the Frontiers of Finite Field Numerical Ranges

Advisor Information

Patrick X. Rault

Location

MBSC Omaha Room 304 - U

Presentation Type

Oral Presentation

Start Date

4-3-2022 10:45 AM

End Date

4-3-2022 12:00 PM

Abstract

Let p be a prime number. There exists a finite version of the complex numbers that have p2 elements, which we call F. Firstly, we will discuss how to compute unit vectors whose coefficients are in F. The numerical range can be understood as a map from the unit sphere to the complex plane F. By using these unit vectors, we will demonstrate plots of the numerical range for various matrices A, and various dimensions n, and primes p. For n > 3 and p > 7, this process is very computationally intensive; consequently, we will discuss the algorithms involved. Lastly, we will explore situations in which the numerical range is the whole set F.

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COinS
 
Mar 4th, 10:45 AM Mar 4th, 12:00 PM

Expanding the Frontiers of Finite Field Numerical Ranges

MBSC Omaha Room 304 - U

Let p be a prime number. There exists a finite version of the complex numbers that have p2 elements, which we call F. Firstly, we will discuss how to compute unit vectors whose coefficients are in F. The numerical range can be understood as a map from the unit sphere to the complex plane F. By using these unit vectors, we will demonstrate plots of the numerical range for various matrices A, and various dimensions n, and primes p. For n > 3 and p > 7, this process is very computationally intensive; consequently, we will discuss the algorithms involved. Lastly, we will explore situations in which the numerical range is the whole set F.