On 21 Jul 2013 21:43:22 GMT, Bart Goddard <email@example.com> wrote:
>firstname.lastname@example.org wrote in news:f6288bc3-b3b4-45bb-9425-2a70f2cea066 >@googlegroups.com: > >> no two polynomials are the same function over finite fields > >I think you mean _infinite_ fields. In which case you can >use functional methods. If the two polynomials give the >same function, then plug 0 in for X to see that their constant >terms are the same. Then take (formal) derivatives and plug >in 0 again to see that the first-order coefficients are the >same, etc.
Not that it matters, since the result is easy to prove otherwise, but I don't see how this works. The problem being that the formal derivative is just "formal":
Looking at the difference of our two polynomials, say p(t) = 0 for all t in our infinite field. So p has zero constant term (hence p(t) = t q(t) for some polynomial t and we're done, hence the "not that it matters" above). How does it follow that p'(t) = 0?
That certainly follows for real or complex polynomials, since the formal derivative is also the derivative, with a definition in terms of the _values_ of the asspciated polynomial function. But..