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Session S09 - Number Theory in the Americas

Thursday, July 15, 13:00 ~ 13:30 UTC-3

Irreducibility of random polynomials of large degree

Dimitris Koukoulopoulos

Let $f(x)=a_0+a_1x+\cdots+a_{n-1}x^{n-1}+x^n$ be a random monic polynomial, where $a_j$ is chosen uniformly at random from $\{0,1\}$ and independently of the other coefficients. In 1993, Odlyzko and Poonen conjectured that $f(x)$ is irreducible with probability $\sim1/2$ when $n\to\infty$. Breuillard and Varj\'u proved that this expectation is indeed true under the Generalized Riemann Hypothesis. In this talk, I will present recent joint work with Bary-Soroker and Kozma that proves that $f(x)$ is irreducible with probability $\ge1/1000$ for all large enough $n$. In addition, if we condition on the event that $f(x)$ is irreducible, then we prove that the Galois group of $f(x)$ contains the alternating group $A_n$ with conditional probability $\sim1$. The proofs use a fun mixture of ideas from sieve methods, the arithmetic of polynomials over finite fields, $p$-adic Fourier analysis, primes with restricted digits, Galois theory and group theory.