Algebra/Number Theory Seminar

Skolem’s question and its variations

Dong Quan Ngoc Nguyen, Notre Dame

Thursday, November 4th, 2021
2:00 PM - 3:00 PM
Zoom id 566 385 6457 (no password)

Skolem, in 1938, asked a question as to whether the group \(SL_n(Z)\) consisting of n by n matrices of determinant 1 with integral entries, admits a polynomial parametrization, that is, whether there exists a matrix polynomial \(A(x_1, \cdots, x_d)\) in \(SL_n(Z[x_1, \cdots, x_d])\) such that every element in \(SL_n(Z)\) can be represented as \(A(a_1, \cdots, a_d)\) for some integers \(a_i\). It was not until 2010 when Vaserstein answered this question in the affirmative. In this talk, I will discuss a generalization of Skolem’s question to other rings including the ring of polynomials over a finite field and number rings. I will also discuss variations of Skolem’s question in the settings of algebraic groups, and of nilpotent groups.

Event contact: dinesh dot thakur at rochester dot edu