Normal limiting distributions for systems of linear equations in random sets
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Juanjo Rué [1] ; Maximilian Wötzel [2]
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[1] Universitat Politècnica de Catalunya
Universitat Politècnica de Catalunya
Barcelona, España
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[2] Radboud University, Nijmegen, The Netherlands
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- Localización: Discrete Mathematics Days 2022 / coord. por Luis Felipe Tabera Alonso, 2022, ISBN 978-84-19024-02-2, págs. 243-248
- Idioma: inglés
- Texto completo no disponible (Saber más ...)
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Resumen
We consider the binomial random set model [n]p where each element in {1, . . . , n} ischosen independently with probability p := p(n). We show that for essentially all regimes ofp and very general conditions for a matrix A and a column vector b, the count of specificinteger solutions to the system of linear equations Ax = b with the entries of x in [n]pfollows a (conveniently rescaled) normal limiting distribution. This applies among othersto the number of solutions with every variable having a different value, as well as to abroader class of so-called non-trivial solutions in homogeneous strictly balanced systems.Our proof relies on the delicate linear algebraic study both of the subjacent matrices andthe corresponding ranks of certain submatrices, together with the application of the methodof moments in probability theory.
