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From: Dave L. Renfro on 20 Jul 2010 14:20
Fred wrote:

> Your paper looks interesting also. It deals with a problem
> that Hamming doesn't address, namely whether something can
> be rational (or algebraic) on some small interval. For example,
> the fact that tan x is not algebraic on its whole domain does
> not say that it couldn't be algebraic on some nonempty open
> interval (although it is not).

One possible avenue for further research would be to
investigate whether such results continue to hold if
"nonempty open interval" is expanded to include all
nonempty sets that are perfect relative to some open
interval. For this, the following paper might be useful,
but this is more a "gut feeling" than a belief based
on my having given this any thought.

Jan Mycielski, "Independent sets in topological algebras",
Fundamenta Mathematicae 55 (1964), 139-147.
http://matwbn.icm.edu.pl/ksiazki/fm/fm55/fm55112.pdf

The following may also be of some marginal relevance:

William F. Donoghue, "Functions which are polynomials
on a dense set", Journal of the London Mathematical
Society 39 (1964), 533-536.

Dave L. Renfro
From: Robert Israel on 20 Jul 2010 16:00
Fred <f.richman(a)comcast.net> writes:

> On Jul 20, 4:29=A0am, Jos=E9 Carlos Santos <jcsan...(a)fc.up.pt> wrote:
> > On 20-07-2010 1:59,Fredwrote:
> >
> > > There's a paper of Hamming in the Monthly in 1970 in which certain
> > > functions, like tan x, are proved to be transcendental. There seem to
> > > be a few gaps in these proofs. I've been told there is a more recent
> > > paper by someone on the same topic in which errors in previous papers
> > > are corrected. Does anyone know of such a paper?
> >
> > I don't know about such paper, but you might be interested in these
> > ones:
> >
> > Jose Carlos Santos and Gabriela Chaves, "Why some elementary functions
> > are not rational" (Mathematics Magazine 77, 2004, 225-226).
> >
> > George P. Speck, "Elementary transcendental functions", Mathematics
> > Magazine 42 (1969), 200-202 (Reprinted on pp. 80-82 of "A Century of
> > Calculus: Part II 1969-1991", The Mathematical Association of America,
> > 1992).
> >
> > Best regards,
> >
> > Jose Carlos Santos
>
> Thanks for the references. The seoond one looks quite relevant. The
> treatment there seems a lot more careful than Hamming's.
>
> Your paper looks interesting also. It deals with a problem that
> Hamming doesn't address, namely whether something can be rational (or
> algebraic) on some small interval. For example, the fact that tan x is
> not algebraic on its whole domain does not say that it couldn't be
> algebraic on some nonempty open interval (although it is not).

This is trivial if you use a little complex analysis.
If f is analytic on a domain D in the complex plane and P is a
polynomial, P(f(z)) is also analytic on D. In particular, if P(f(z)) = 0 for
z in some interval contained in D, then P(f(z)) = 0 everywhere in D .
Thus an analytic function can't be algebraic on part of its domain but not on
the whole domain. For example, tan, which is analytic on the complex plane
except for poles at n pi/2 for odd integers n, can't be algebraic on
any interval.
--
Robert Israel israel(a)math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
From: Dave L. Renfro on 20 Jul 2010 16:33
Robert Israel

> This is trivial if you use a little complex analysis.
> If f is analytic on a domain D in the complex plane and P
> is a polynomial, P(f(z)) is also analytic on D. In particular,
> if P(f(z)) = 0 for z in some interval contained in D, then
> P(f(z)) = 0 everywhere in D . Thus an analytic function can't
> be algebraic on part of its domain but not on the whole domain.
> For example, tan, which is analytic on the complex plane except
> for poles at n pi/2 for odd integers n, can't be algebraic on
> any interval.

I was wondering if the uniqueness result for complex analytic
functions made all this a moot point (including the perfect set
version I posted about earlier today), but I figured there must
be something blocking its use since I don't believe it's mentioned
in any of the papers on the subject (of trig. functions not being
algebraic functions). I meant to ask about this in the post I recently
made (about the perfect set version -- which is also taken care of
by the uniqueness result) and forgot, which is kind of strange
since the uniqueness result possibly playing a role didn't occur
to me until I began wondering about the perfect set version.

Dave L. Renfro
From: Fred on 20 Jul 2010 20:03
On Jul 20, 4:00 pm, Robert Israel
<isr...(a)math.MyUniversitysInitials.ca> wrote:
> Fred <f.rich...(a)comcast.net> writes:
> > On Jul 20, 4:29=A0am, Jos=E9 Carlos Santos <jcsan...(a)fc.up.pt> wrote:
> > > On 20-07-2010 1:59,Fredwrote:
>
> > > > There's a paper of Hamming in the Monthly in 1970 in which certain
> > > > functions, like tan x, are proved to be transcendental. There seem to
> > > > be a few gaps in these proofs. I've been told there is a more recent
> > > > paper by someone on the same topic in which errors in previous papers
> > > > are corrected. Does anyone know of such a paper?
>
> > > I don't know about such paper, but you might be interested in these
> > > ones:
>
> > > Jose Carlos Santos and Gabriela Chaves, "Why some elementary functions
> > > are not rational" (Mathematics Magazine 77, 2004, 225-226).
>
> > > George P. Speck, "Elementary transcendental functions", Mathematics
> > > Magazine 42 (1969), 200-202 (Reprinted on pp. 80-82 of "A Century of
> > > Calculus: Part II 1969-1991", The Mathematical Association of America,
> > > 1992).
>
> > > Best regards,
>
> > > Jose Carlos Santos
>
> > Thanks for the references. The seoond one looks quite relevant. The
> > treatment there seems a lot more careful than Hamming's.
>
> > Your paper looks interesting also. It deals with a problem that
> > Hamming doesn't address, namely whether something can be rational (or
> > algebraic) on some small interval. For example, the fact that tan x is
> > not algebraic on its whole domain does not say that it couldn't be
> > algebraic on some nonempty open interval (although it is not).
>
> This is trivial if you use a little complex analysis.
> If f is analytic on a domain D in the complex plane and P is a
> polynomial, P(f(z)) is also analytic on D.  In particular, if P(f(z)) = 0 for
> z in some interval contained in D, then P(f(z)) = 0 everywhere in D .
> Thus an analytic function can't be algebraic on part of its domain but not on
> the whole domain.  For example, tan, which is analytic on the complex plane
> except for poles at n pi/2 for odd integers n, can't be algebraic on
> any interval.
> --
> Robert Israel              isr...(a)math.MyUniversitysInitials.ca
> Department of Mathematics        http://www.math.ubc.ca/~israel
> University of British Columbia            Vancouver, BC, Canada


That's pretty neat, I didn't think of going complex. I was fixated on
the reals where you can't use that argument because the domain is
disconnected.

--Fred
From: G. A. Edgar on 21 Jul 2010 06:32
In article <rbisrael.20100720192951$4b6f(a)news.acm.uiuc.edu>, Robert
Israel <israel(a)math.MyUniversitysInitials.ca> wrote:

> [...]
> If f is analytic on a domain D in the complex plane and P is a
> polynomial, P(f(z)) is also analytic on D. In particular, if P(f(z)) = 0 for
> z in some interval contained in D, then P(f(z)) = 0 everywhere in D .

Maybe better if you say P(z,f(z)) in place of P(z) here. But it still
works.

A remark on the definition of "algebraic function" ... If we are not
starting with "analytic function f in a domain of the complex plane",
then what are we starting with? For example, we do not want to allow:
f(x) = sqrt(1-x^2) for x rational in (-1,1) and f(x) = -sqrt(1-x^2)
for x irrational in (-1,1).

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
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