[math-fun] Integers, rationals, algebraics, ?
An algebraic number of degree d is a root of some degree d integer polynomial that is irreducible (not the product of two polynomials of positive degree). Question 1: ----------- What is the set of non-algebraic roots of analytic functions with integer coefficients? —Dan
What do you mean by the "coefficients" of an analytic function? The coefficients of its power series centered at the origin? Such a series, unless it has only finitely many terms (in which case its roots are algebraic0, will diverge at 1, so one ting you can say about the set is "it's contained in the unit circle" I'm not sure what you mean by "what *is* the set of...". I don't think this set will have any simpler or more familiar description. It might be more fruitful to ask about some particular properties of the set. For example, I'm pretty sure it's uncountable, but don't have an immediate proof. Andy On Sat, Feb 10, 2018 at 4:16 PM, Dan Asimov <dasimov@earthlink.net> wrote:
An algebraic number of degree d is a root of some degree d integer polynomial that is irreducible (not the product of two polynomials of positive degree).
Question 1: ----------- What is the set of non-algebraic roots of analytic functions with integer coefficients?
—Dan
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-- Andy.Latto@pobox.com
I'm not sure what you mean by "what *is* the set of...". I don't think this set will have any simpler or more familiar description. It might be more fruitful to ask about some particular properties of the set. For example, I'm pretty sure it's uncountable, but don't have an immediate proof.
I suspect it's the open unit disc. A proof would proceed along the following lines: (a) Consider the space X of power series with integer coefficients and positive radius of convergence as a subset of Baire space: https://en.wikipedia.org/wiki/Baire_space_(set_theory) (b) Show that if a sequence of points in X converges, then the roots of the series(es) also converge. (c) Show that for each compact subset K of the unit disc, we can find a compact subset of X whose image contains a dense subset of K (and therefore contains the whole of K by part (b)). Best wishes, Adam P. Goucher
Andy
On Sat, Feb 10, 2018 at 4:16 PM, Dan Asimov <dasimov@earthlink.net> wrote:
An algebraic number of degree d is a root of some degree d integer polynomial that is irreducible (not the product of two polynomials of positive degree).
Question 1: ----------- What is the set of non-algebraic roots of analytic functions with integer coefficients?
—Dan
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-- Andy.Latto@pobox.com
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participants (3)
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Adam P. Goucher -
Andy Latto -
Dan Asimov