Full Galois groups of polynomials with slowly growing coefficients

2024/04/21 by Bary-Soroker, Lior, Goldgraber, Noam · 2 citations
#11R09 #11R32 #11T23 #11T55 #12F10 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2404.13559

Abstract

Choose a polynomial f uniformly at random from the set of all monic polynomials of degree n with integer coefficients in the box [-L,L]ⁿ. The main result of the paper asserts that if L=L(n) grows to infinity, then the Galois group of f is the full symmetric group, asymptotically almost surely, as n→ ∞. When L grows rapidly to infinity, say L>n⁷, this theorem follows from a result of Gallagher. When L is bounded, the analog of the theorem is open, while the state-of-the-art is that the Galois group is large in the sense that it contains the alternating group (if L< 17, it is conditional on the general Riemann hypothesis). Hence the most interesting case of the theorem is when L grows slowly to infinity. Our method works for more general independent coefficients.

Cited by

• Irreducibility and Galois groups of random reciprocal polynomials of large degree
• Law of large numbers for the discriminant of random polynomials

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