# Thread: prove f is a rational polynomial

1. ## prove f is a rational polynomial

$f(x)$ is polynomial with complex coefficients. $\forall n\in Z$, $f(n)$is integer, prove: coefficients of $f(x)$ are rational numbers, and give some examples about rational case.

### Prove:

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* consider coefficients are integers, of course $f(n)$ are integers.

* consider coefficients are rationals, we have $f(x)=\frac{1}{2}x(x+1)$, two consecutive integer can be divided by $2$, there must be one even number.

* real coeffs

* complex coeffs

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And can you give me some more examples?

2. ## Re: prove f is a rational polynomial

The following ideas are based on this article (PDF). It turns out that every integer-valued polynomial p(x) has the form

$p(x) = a_0+a_1\binom{x}{1}+ \dots+ a_n\binom{x}{n}$ (*)

Indeed, the coefficients (omitting the constant term, which is zero) of polynomials $\binom{x}{1}$, $\binom{x}{2}$, ..., $\binom{x}{n}$ form a triangular matrix, so these polynomials form a basis, i.e., every polynomial of degree ≤ n with no constant term can be expressed as a linear combination of these polynomials, perhaps with complex coefficients. So, every polynomials from ℂ[x] of degree ≤ n can be written in the form (*) with complex $a_k$. Next prove by induction on k that $a_k\in\mathbb{Z}$.