Computing Generalized Chebyshev Polynomials and Associated Finite Field Parametrizations

Date

2025-8-26

Author

Azmaz, Metin

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Let $\mf{g}$ be a semisimple Lie algebra. The Weyl group of $\mf{g}$ acts on the lattice generated by the fundamental weights, and the exponential orbit sums associated with these weights form invariants under the group. Generalized Chebyshev polynomials $P^K_\mf{g}$ are defined via these exponential invariants, where $K$ is an integer matrix. In this thesis, we develop algorithms in the PARI/GP programming language to compute Weyl groups, generalized Chebyshev polynomials $P^K_\mf{g}$, and parametrizations of $\F_q^n$ using fixed points of these polynomials. We also analyze the value sets of these polynomials over finite fields and investigate certain properties such as arithmetic exceptionality.

Subject Keywords

Lie algebras, Weyl groups, exponential invariants, PARI/GP, computational algebra

URI

https://hdl.handle.net/11511/115681

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Graduate School of Natural and Applied Sciences, Thesis