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Post Closed as "Not suitable for this site" by Bill Johnson, Felipe Voloch, Qiaochu Yuan, Neil Strickland, Alex Degtyarev
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Prove What conditions imply that a function over $\mathbb{Z}$ is a polynomial?

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Halbort
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How would one prove that a function is a polynomial? I can't seem to find anything about this on the internet. I would like to know if there are any unique properties that only polynomials can satisfy. Given properties of a function, can these properties be used to prove that a function is a polynomial. I would like to do this without actually constructing the polynomial. What I mean is something like this: One can show that the sine function is not a polynomial without deriving an actual formula. You just use the property that a polynomial has finite degree.

EDIT: My function is only defined on the positive integers. So I am not sure how to use the ideas of complex analysis.

How would one prove that a function is a polynomial? I can't seem to find anything about this on the internet. I would like to know if there are any unique properties that only polynomials can satisfy. Given properties of a function, can these properties be used to prove that a function is a polynomial. I would like to do this without actually constructing the polynomial. What I mean is something like this: One can show that the sine function is not a polynomial without deriving an actual formula. You just use the property that a polynomial has finite degree.

How would one prove that a function is a polynomial? I can't seem to find anything about this on the internet. I would like to know if there are any unique properties that only polynomials can satisfy. Given properties of a function, can these properties be used to prove that a function is a polynomial. I would like to do this without actually constructing the polynomial. What I mean is something like this: One can show that the sine function is not a polynomial without deriving an actual formula. You just use the property that a polynomial has finite degree.

EDIT: My function is only defined on the positive integers. So I am not sure how to use the ideas of complex analysis.

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Halbort
Halbort
  • 1.1k
  • 1
  • 7
  • 20

Prove a function is polynomial?

How would one prove that a function is a polynomial? I can't seem to find anything about this on the internet. I would like to know if there are any unique properties that only polynomials can satisfy. Given properties of a function, can these properties be used to prove that a function is a polynomial. I would like to do this without actually constructing the polynomial. What I mean is something like this: One can show that the sine function is not a polynomial without deriving an actual formula. You just use the property that a polynomial has finite degree.