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A polynomial over the reals is an infinite sequence of real numbers, the coefficients, of which only finitely many are different from 0:
p is polynomial : <=> p: N -> R /\ (exists k in N: forall i >= k: pi = 0).
The degree of a polynomial is zero, if all coefficients are zero; otherwise, it is the index of the largest non-zero coeffient:
deg(p) := if forall i in N: pi = 0 then 0 else (such k in N: pk != 0 /\ (forall i > k: pi = 0)).