Galois groups of polynomials of large degree

Abstract: Abstract: Let $\mathcal{N}$ be a set of natural numbers and let us consider all monic polynomials of degree $n$ whose coefficients are in $\mathcal{N}$. What are the odds that a polynomial chosen uniformly at random from this set is irreducible? Moreover, what can we say about the distribution of its Galois group? I will present some recent results, joint with Lior Bary-Soroker and Gady Kozma, that show that if $\mathcal{N}$ is not too sparse, then such random polynomials are highly likely to be irreducible and have very large Galois group. The proofs uses a fun mixture of ideas from sieve methods, the arithmetic of polynomials over finite fields, primes with restricted digits, Galois theory and group theory.

combinatoricsnumber theory

Audience: researchers in the topic

Webinar in Additive Combinatorics

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