Re: Can someone explain this proof, please?

victor_meldrew_666@xxxxxxxxxxx wrote:

Timothy Murphy wrote:

I take it that your proof shows that if the solution is written
f(x) g(x) = C(x),
then we get the same solution on replacing x by 1/x
ie (x^n f(1/x)) (x^m g(1/x)) = x^{m+n} C(1/x) = C(x).

An easy way to see that f and g are reciprocal polynomials
is to note that each zero z of f is a root of unity
and so has modulus 1. As f has real coefficients the
conjugate of z which is z^{-1} is also a root. Hence
f is reciprocal.

So the second part of your Hypothesis H(j) -
that a_i = a_{n-i} and b_i = b_{m-i}
could have been proved first, as a lemma.
The inductive hypothesis would then simply have been
that a_i,b_i are 0 or 1, which is a very natural hypothesis.

I'm coming round to the view that the problem might have been
easier than I thought, though hardly "trivial".

--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
.

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