{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,7]],"date-time":"2023-11-07T05:15:23Z","timestamp":1699334123164},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2021,8,7]],"date-time":"2021-08-07T00:00:00Z","timestamp":1628294400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,8,7]],"date-time":"2021-08-07T00:00:00Z","timestamp":1628294400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"University of Helsinki including Helsinki University Central Hospital"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Period Math Hung"],"published-print":{"date-parts":[[2022,6]]},"abstract":"Abstract<\/jats:title>In this paper, we study how the roots of the Kac polynomials$$W_n(z) = \\sum _{k=0}^{n-1} \\xi _k z^k$$<\/jats:tex-math>W<\/mml:mi>n<\/mml:mi><\/mml:msub>(<\/mml:mo>z<\/mml:mi>)<\/mml:mo><\/mml:mrow>=<\/mml:mo>\u2211<\/mml:mo>k<\/mml:mi>=<\/mml:mo>0<\/mml:mn><\/mml:mrow>n<\/mml:mi>-<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:msubsup>\u03be<\/mml:mi>k<\/mml:mi><\/mml:msub>z<\/mml:mi>k<\/mml:mi><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>concentrate around the unit circle when the coefficients of$$W_n$$<\/jats:tex-math>W<\/mml:mi>n<\/mml:mi><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>are independent and identically distributed nondegenerate real random variables. It is well known that the roots of a Kac polynomial concentrate around the unit circle as$$n\\rightarrow \\infty $$<\/jats:tex-math>n<\/mml:mi>\u2192<\/mml:mo>\u221e<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>if and only if$${\\mathbb {E}}[\\log ( 1+ |\\xi _0|)]<\\infty $$<\/jats:tex-math>E<\/mml:mi>[<\/mml:mo>log<\/mml:mo>(<\/mml:mo>1<\/mml:mn>+<\/mml:mo>|<\/mml:mo><\/mml:mrow>\u03be<\/mml:mi>0<\/mml:mn><\/mml:msub>|<\/mml:mo>)<\/mml:mo>]<\/mml:mo><<\/mml:mo>\u221e<\/mml:mi><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Under the condition$${\\mathbb {E}}[\\xi ^2_0]<\\infty $$<\/jats:tex-math>E<\/mml:mi>[<\/mml:mo>\u03be<\/mml:mi>0<\/mml:mn>2<\/mml:mn><\/mml:msubsup>]<\/mml:mo><<\/mml:mo>\u221e<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we show that there exists an annulus of width$${\\text {O}}(n^{-2}(\\log n)^{-3})$$<\/jats:tex-math>O<\/mml:mtext>(<\/mml:mo>n<\/mml:mi>-<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:msup>(<\/mml:mo>log<\/mml:mo>n<\/mml:mi>)<\/mml:mo><\/mml:mrow>-<\/mml:mo>3<\/mml:mn><\/mml:mrow><\/mml:msup>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>around the unit circle which isfree<\/jats:italic>of roots with probability$$1-{\\text {O}}({(\\log n)^{-{1}\/{2}}})$$<\/jats:tex-math>1<\/mml:mn>-<\/mml:mo>O<\/mml:mtext>(<\/mml:mo>(<\/mml:mo>log<\/mml:mo>n<\/mml:mi>)<\/mml:mo><\/mml:mrow>-<\/mml:mo>1<\/mml:mn>\/<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:msup>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The proof relies on small ball probability inequalities and the least common denominator used in [17].<\/jats:p>","DOI":"10.1007\/s10998-021-00409-7","type":"journal-article","created":{"date-parts":[[2021,8,7]],"date-time":"2021-08-07T16:02:43Z","timestamp":1628352163000},"page":"159-176","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Zero-free neighborhoods around the unit circle for Kac polynomials"],"prefix":"10.1007","volume":"84","author":[{"given":"Gerardo","family":"Barrera","sequence":"first","affiliation":[]},{"given":"Paulo","family":"Manrique","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,8,7]]},"reference":[{"key":"409_CR1","doi-asserted-by":"crossref","unstructured":"T. Apostol, Introduction to analytic number theory. 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