Re: Attacking my algebraic integer work

From: William Hughes (wpihughes_at_hotmail.com)
Date: 03/29/05 

Date: 28 Mar 2005 17:30:05 -0800

jstevh@msn.com wrote:
> William Hughes wrote:
> > jstevh@msn.com wrote:
> > > William Hughes wrote:
> >
> > <snip>
> >
> > > >
> > > > A key part of your arguments has been that dividing through
> > > > by a non-constant factor cannot lead to a counterexample as
> > > > this would involve changing the value of the constant term. In
> > this
> > >
> > > I'm not sure about your claim here about that being a key part of
> my
> > > argument.
> > >
> > > > context it has been noted that the question
> > > >
> > > > Given a(0) = 0, w(x) non-constant and w(0) = 1, what
> > > > is the constant term of
> > > >
> > > > g(x) = a(x)/w(x) + 7/w(x) ?
> > > >
> > > > is critical . However, you have ignored this question.
> > >
> > > If g(x) is a continuous function, lies in the ring of algebraic
> > > integers, and w(x) is a continuous function in that ring, which
is
> a
> > > factor of g(x) in that ring, then
> > >
> > > g(0) = 7.
> > >
> > > I use the ring of algebraic integers here as it's simpler.
> > >
> > > You can move outside of that ring to consider constant terms, but
> > I'll
> > > start simply and see how it goes.
> > >
> >
> >
> > Are you now saying that the constant terms of
> >
> > g(x) = a(x)/w(x) + 7/w(x)
> >
> > and
> >
> > h(x) = a(x) + 7
> >
> > are the same? If so, why do you think that dividing by a
> non-constant
> > w(x) changes the constant term?
> >
>
> It seems to me that for the purposes of carefully considering what
you
> are trying to say, it'd be good to go slowly, and step through
> everything.
>
> No assumptions. Don't leave something out thinking it's just
> understood.
>
> I'm not "now saying" anything as if I were saying something else
> before, as you yourself even claimed I was avoiding the question
before
> (not that I'm saying I was as I don't remember or care), but now
you're
> saying that I'm now saying as if I had some other position before?
>
> My position is that constant terms are those familiar constants that
> people see with an expression like
>
> f(x) = x + 2
>
> where 2 is the constant term, or with
>
> f(x) = x^2 + 5x + 6
>
> where 6 is the constant term.
>
> At times I've abstracted to using letters in place of numbers, for
> instance
>
> f(x) = x + c
>
> where c is a constant, independent of x, as being independent of x is
> what makes it a constant term, and often to make that certain I'd say
> constant with respect to x, but when posters found that problematic,
I
> could stick in numbers and my favorite has been
>
> 1078
>
> which is 7(7)(22).
>
> That is a constant.
>
> And that's what I mean when I talk about constant terms.
>
> What did you think I meant?
>

My understanding is that you are currently using the definition
"for any function f(x), the constant term of f is f(0)"

                    - "William Hughes"

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