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Algebra, matrices, and computers (2019)
Journal Article
Detinko, A., Flannery, D., & Hulpke, A. (2019). Algebra, matrices, and computers. Snapshots of modern mathematics from Oberwolfach, 5, 1-12. https://doi.org/10.14760/SNAP-2019-005-EN

What part does algebra play in representing the real world abstractly? How can algebra be used to solve hard mathematical problems with the aid of modern computing technology? We provide answers to these questions that rely on the theory of matrix gr... Read More

The strong approximation theorem and computing with linear groups (2019)
Journal Article
Detinko, A., Flannery, D., & Hulpke, A. (2019). The strong approximation theorem and computing with linear groups. Journal of Algebra, 529, 536-549. https://doi.org/10.1016/j.jalgebra.2019.04.011

We obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group for . More generally, we are able to com... Read More

Practical computation with linear groups over infinite domains (2019)
Book Chapter
Detinko, A. S., & Flannery, D. L. (2019). Practical computation with linear groups over infinite domains. In C. Campbell, C. Parker, M. Quick, E. Robertson, & C. Roney-Dougal (Eds.), Groups St Andrews 2017 in Birmingham, 261-270. Cambridge University Press (CUP). https://doi.org/10.1017/9781108692397.011

We survey recent progress in computing with finitely generated linear groups over infinite fields, describing the mathematical background of a methodology applied to design practical algorithms for these groups. Implementations of the algorithms have... Read More

Linear groups and computation (2018)
Journal Article
Detinko, A. S., & Flannery, D. L. (in press). Linear groups and computation. Expositiones Mathematicae, https://doi.org/10.1016/j.exmath.2018.07.002

We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for this class of groups are surveyed. We... Read More

Algorithms for experimenting with Zariski dense subgroups (2018)
Journal Article
Detinko, A. S., Flannery, D. L., & Hulpke, A. (in press). Algorithms for experimenting with Zariski dense subgroups. Experimental Mathematics, https://doi.org/10.1080/10586458.2018.1466217

We give a method to describe all congruence images of a finitely generated Zariski dense group . The method is applied to obtain efficient algorithms for solving this problem in odd prime degree n; if n = 2 then we compute all congruence images only... Read More

Zariski density and computing in arithmetic groups (2017)
Journal Article
Detinko, A., Flannery, D. L., & Hulpke, A. (2017). Zariski density and computing in arithmetic groups. Mathematics of Computation, 87(310), 967-986. https://doi.org/10.1090/mcom/3236

For n > 2, let Gamma _n denote either SL(n, {Z}) or Sp(n, {Z}). We give a practical algorithm to compute the level of the maximal principal congruence subgroup in an arithmetic group H\leq \Gamma _n. This forms the main component of our methods for c... Read More

Integrality and arithmeticity of solvable linear groups (2014)
Journal Article
Detinko, A., Flannery, D., & de Graaf, W. (2015). Integrality and arithmeticity of solvable linear groups. Journal of Symbolic Computation, 68(Part 1), 138-145. https://doi.org/10.1016/j.jsc.2014.08.011

We develop a practical algorithm to decide whether a finitely generated subgroup of a solvable algebraic group G is arithmetic. This incorporates a procedure to compute a generating set of an arithmetic subgroup of G. We also provide a simple new alg... Read More