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• W2605934764 abstract "Let<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K><mml:semantics><mml:mi>K</mml:mi><mml:annotation encoding=application/x-tex>K</mml:annotation></mml:semantics></mml:math></inline-formula>be a number field with ring of integers<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper O Subscript upper K><mml:semantics><mml:msub><mml:mrow class=MJX-TeXAtom-ORD><mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msub><mml:annotation encoding=application/x-tex>mathcal {O}_K</mml:annotation></mml:semantics></mml:math></inline-formula>, and let<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-brace f Subscript k Baseline right-brace Subscript k element-of double-struck upper N><mml:semantics><mml:mrow><mml:mo fence=false stretchy=false>{</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msub><mml:mo fence=false stretchy=false>}</mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>k</mml:mi><mml:mo>∈<!-- ∈ --></mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=double-struck>N</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mrow><mml:annotation encoding=application/x-tex>{f_k}_{kin mathbb {N}}</mml:annotation></mml:semantics></mml:math></inline-formula>be a sequence of monic polynomials in<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper O Subscript upper K Baseline left-bracket x right-bracket><mml:semantics><mml:mrow><mml:msub><mml:mrow class=MJX-TeXAtom-ORD><mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msub><mml:mo stretchy=false>[</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=false>]</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>mathcal {O}_K[x]</mml:annotation></mml:semantics></mml:math></inline-formula>such that for every<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n element-of double-struck upper N><mml:semantics><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈<!-- ∈ --></mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=double-struck>N</mml:mi></mml:mrow></mml:mrow><mml:annotation encoding=application/x-tex>nin mathbb {N}</mml:annotation></mml:semantics></mml:math></inline-formula>, the composition<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f Superscript left-parenthesis n right-parenthesis Baseline equals f 1 ring f 2 ring ellipsis ring f Subscript n><mml:semantics><mml:mrow><mml:msup><mml:mi>f</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mo stretchy=false>(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>∘<!-- ∘ --></mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>∘<!-- ∘ --></mml:mo><mml:mo>…<!-- … --></mml:mo><mml:mo>∘<!-- ∘ --></mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:annotation encoding=application/x-tex>f^{(n)}=f_1circ f_2circ ldots circ f_n</mml:annotation></mml:semantics></mml:math></inline-formula>is irreducible. In this paper we show that if the size of the Galois group of<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f Superscript left-parenthesis n right-parenthesis><mml:semantics><mml:msup><mml:mi>f</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mo stretchy=false>(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:mrow></mml:msup><mml:annotation encoding=application/x-tex>f^{(n)}</mml:annotation></mml:semantics></mml:math></inline-formula>is large enough (in a precise sense) as a function of<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n><mml:semantics><mml:mi>n</mml:mi><mml:annotation encoding=application/x-tex>n</mml:annotation></mml:semantics></mml:math></inline-formula>, then the set of primes<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=German p subset-of-or-equal-to script upper O Subscript upper K><mml:semantics><mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=fraktur>p</mml:mi></mml:mrow><mml:mo>⊆<!-- ⊆ --></mml:mo><mml:msub><mml:mrow class=MJX-TeXAtom-ORD><mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msub></mml:mrow><mml:annotation encoding=application/x-tex>mathfrak {p}subseteq mathcal {O}_K</mml:annotation></mml:semantics></mml:math></inline-formula>such that every<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f Superscript left-parenthesis n right-parenthesis><mml:semantics><mml:msup><mml:mi>f</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mo stretchy=false>(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:mrow></mml:msup><mml:annotation encoding=application/x-tex>f^{(n)}</mml:annotation></mml:semantics></mml:math></inline-formula>is irreducible modulo<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=German p><mml:semantics><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=fraktur>p</mml:mi></mml:mrow><mml:annotation encoding=application/x-tex>mathfrak {p}</mml:annotation></mml:semantics></mml:math></inline-formula>has density zero. Moreover, we prove that the subset of polynomial sequences such that the Galois group of<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f Superscript left-parenthesis n right-parenthesis><mml:semantics><mml:msup><mml:mi>f</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mo stretchy=false>(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:mrow></mml:msup><mml:annotation encoding=application/x-tex>f^{(n)}</mml:annotation></mml:semantics></mml:math></inline-formula>is large enough has density 1, in an appropriate sense, within the set of all polynomial sequences." @default.
• W2605934764 created "2017-04-28" @default.
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• W2605934764 date "2018-02-16" @default.
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• W2605934764 title "The set of stable primes for polynomial sequences with large Galois group" @default.
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• W2605934764 doi "https://doi.org/10.1090/proc/13958" @default.
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