Ilaria Viglino: Almost sure convergence of least common multiple of ideals for polynomials over a number field

Date: 2024-02-21

Time: 14:10 - 15:00

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Ilaria Viglino, EPFL


For \( f \in \mathbb{Z}[X] \) an irreducible polynomial of degree \( n \), the Cilleruelo’s conjecture states
\log(\text{lcm}(f(1),\dots,f(M))) \sim (n – 1)M \log M 
as \( M \rightarrow +\infty \). The Prime Number Theorem for arithmetic progressions can be exploited to obtain an asymptotic estimate when \( f \) is a linear polynomial. Cilleruelo extended this result to quadratic polynomials. The asymptotic remains unknown for irreducible polynomials of higher degree. Recently the conjecture was shown on average for a large family of polynomials of any degree by Rudnick and Zehavi. We investigate the case of \( S_n \)-polynomials with coefficients in the ring of algebraic integers of a fixed number field extension \( K/\mathbb{Q} \) by considering the least common multiple of ideals of \( \mathcal{O}_K \).