Factorization in Fq[x] and Brownian Motion
- 12 September 1993
- journal article
- research article
- Published by Cambridge University Press (CUP) in Combinatorics, Probability and Computing
- Vol. 2 (3) , 285-299
- https://doi.org/10.1017/s0963548300000687
Abstract
We consider the set of polynomials of degree n over a finite field and put the uniform probability measure on this set. Any such polynomial factors uniquely into a product of its irreducible factors. To each polynomial we associate a step function on the interval [0,1] such that the size of each jump corresponds to the number of factors of a certain degree in the factorization of the random polynomial. We normalize these random functions and show that the resulting random process converges weakly to Brownian motion as n → ∞. This result complements earlier work by the author on the order statistics of the degree sequence of the factors of a random polynomial.Keywords
This publication has 11 references indexed in Scilit:
- How random is the characteristic polynomial of a random matrix ?Mathematical Proceedings of the Cambridge Philosophical Society, 1993
- On random polynomials over finite fieldsMathematical Proceedings of the Cambridge Philosophical Society, 1993
- Random Matrices and Brownian MotionCombinatorics, Probability and Computing, 1993
- Limit Theorems for Combinatorial Structures via Discrete Process ApproximationsRandom Structures & Algorithms, 1992
- Gaussian limiting distributions for the number of components in combinatorial structuresJournal of Combinatorial Theory, Series A, 1990
- A functional central limit theorem for the Ewens sampling formulaJournal of Applied Probability, 1990
- A Functional Central Limit Theorem for Random MappingsThe Annals of Probability, 1989
- Random permutations and Brownian motionPacific Journal of Mathematics, 1985
- Ordered Cycle Lengths in a Random PermutationTransactions of the American Mathematical Society, 1966
- Ordered cycle lengths in a random permutationTransactions of the American Mathematical Society, 1966