Integer-valued rational polynomials
Which polynomials f(x) in Q[x] have the property that f(n) is
an integer for all integers n?
- For each non-negative integer j, define pj(x) in
Q[x] by
|
pj(x) = |
x (x-1) (x-2) ... (x-j+1)
j!
|
. |
|
(When j=0, p0(x) = 1.)
Show that any polynomial in Q[x] can be written as a
rational linear combination of the polynomials pj(x).
- Let I be this set of rational polynomials:
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I = {f(x) in Q[x] : f(n) is in
Z for all n in
Z
}. |
|
Show that f(x) is in I if and only if f(x) is an integer linear
combination of the polynomials pj(x). [Hint: if f(x) is in
I, evaluate f(x) at the integers 0, 1, ... k.]
(This is from D'Angelo and West, Mathematical Thinking, 2nd
edition, problem 5.65.)
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On 7 Feb 2005, 13:26.
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