Mathematics Colloquia and Seminars

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Starting with the work of Littlewood, Offord and Kac in the 1930s and 1940s, there is now a substantial literature on the number and location of real zeros of polynomials with random real coefficients. Using an idea of Kurt Mahler, I will consider similar questions for random polynomials over other fields such as the p-adics and find simple, non-asymptotic expressions for the expected number of simultaneous zeros of several natural systems of such polynomials. A crucial ingredient in the proof is a theorem on the distribution of the so-called elementary divisors of random matrices with entries from fields such as the p-adics. This latter result uses some observations of Richard Brent and Brendan McKay and has connections with Jason Fulman's probabilistic proof of the Rogers-Ramanujan identities. I will not assume any prior knowledge about the p-adics or local fields in general.