Are Polynomials really C ^Infinity (Smooth) Differentiable

Simplistically, one would expect a Polynomial of degree n to be
differentiable only (n+1) times before arriving at zero. Yet, according to
the public math authorities (Mathworld and Wikipedia) Polynomials are
C^Infinity (smooth); i.e. can be differentiated an unlimited number of times
= continuously differentiable.

Quote from Mathworld on C^Infinity

"A C^Inf function is a function that is differentiable for all degrees of
differentiation. .....(snip) .... All polynomials are . "

Quote from Wkiipedia on polynomials

"Polynomial functions are an important class of smooth functions; smooth
meaning that they are infinitely differentiable, i.e., they have derivatives
of all finite orders."

Why are Polynomials C^Infinity ? Is it only a convenient definition ? If
so, how is it justified ? Can zero really be differentiated continuously ?

Can zero be differentiated at all ? if so, what is its differential ?

References would be appreciated.



.

Relevant Pages

  • Re: Solving overdetermined equations
    ... I rewrote the polynomials in syntax that most math programs will ... Since you wish to avoid solutions where either variable is zero, ... To get an exact Groebner basis I found it expedient to take two steps. ...
    (sci.math.symbolic)
  • Re: Solving overdetermined equations
    ... I rewrote the polynomials in syntax that most math programs will ... Since you wish to avoid solutions where either variable is zero, ... To get an exact Groebner basis I found it expedient to take two steps. ...
    (sci.math.research)
  • Re: Solving overdetermined equations
    ... >> Since you wish to avoid solutions where either variable is zero, ... >> no nontrivial solutions. ... nontrivial polynomials that ultimately are eliminated by later ones. ... Daniel Lichtblau ...
    (sci.math.symbolic)
  • Polynomial root-finding
    ... the art" of certain polynomial root finders. ... I'm writing a paper about polynomials and the process of zero finding ... Needs a polynomial zero finder itself (matlab built in roots function). ...
    (sci.math.num-analysis)
  • Re: Moving on to real math - and help with an exercise
    ... a_j have to be zero, and for part b), that all the ... between the polynomials has to be zero for each ... screaming denial of self-worth sets in, ... for your life, if you haven't started having this ...
    (sci.math)