Where are the zeroes of a random p-adic polynomial?
Forum of Mathematics, Sigma, Tome 10 (2022)
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We study the distribution of the roots of a random p-adic polynomial in an algebraic closure of ${\mathbb Q}_p$. We prove that the mean number of roots generating a fixed finite extension K of ${\mathbb Q}_p$ depends mostly on the discriminant of K, an extension containing fewer roots when it becomes more ramified. We prove further that for any positive integer r, a random p-adic polynomial of sufficiently large degree has about r roots on average in extensions of degree at most r.Beyond the mean, we also study higher moments and correlations between the number of roots in two given subsets of ${\mathbb Q}_p$ (or, more generally, of a finite extension of ${\mathbb Q}_p$). In this perspective, we notably establish results highlighting that the roots tend to repel each other and quantify this phenomenon.
@article{10_1017_fms_2022_27,
author = {Xavier Caruso},
title = {Where are the zeroes of a random p-adic polynomial?},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {10},
year = {2022},
doi = {10.1017/fms.2022.27},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2022.27/}
}
Xavier Caruso. Where are the zeroes of a random p-adic polynomial?. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.27
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