Probabilistic Galois theory in function fields
We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial f=yn+∑i=0n−1ai(x)yi∈Fq[x][y] with i.i.d. coefficients ai taking values in the set {a(x)∈Fq[x]:dega≤d} with uniform probability, is irreducible with probability tending to 1−1qd...
Gespeichert in:
Veröffentlicht in: | Finite fields and their applications 2024-09, Vol.98, p.102466, Article 102466 |
---|---|
Hauptverfasser: | , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext |
Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
Zusammenfassung: | We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial f=yn+∑i=0n−1ai(x)yi∈Fq[x][y] with i.i.d. coefficients ai taking values in the set {a(x)∈Fq[x]:dega≤d} with uniform probability, is irreducible with probability tending to 1−1qd as n→∞, where d and q are fixed. We also prove that with the same probability, the Galois group of this random polynomial contains the alternating group An. Moreover, we prove that if we assume a version of the polynomial Chowla conjecture over Fq[x], then the Galois group of this polynomial is actually equal to the symmetric group Sn with probability tending to 1−1qd. We also study the other possible Galois groups occurring with positive limit probability. Finally, we study the same problems with n fixed and d→∞. |
---|---|
ISSN: | 1071-5797 1090-2465 |
DOI: | 10.1016/j.ffa.2024.102466 |