Find all polynomials which satisfy the identity

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- July 17th 2009, 12:36 AMNonCommAlgA functional equation
Find all polynomials which satisfy the identity

- July 17th 2009, 02:12 AMTwistedOne151
, so if , we see that

.

--Kevin C. - July 18th 2009, 10:37 PMNonCommAlg
- July 30th 2009, 06:11 PMhalbard
OK, let's offer this to the snake and see if it strikes.

Using the functional equation, first show that if is a root of then so are and .

Also, since is a polynomial equation it has only a finite number of roots.

Now let . By the previous assertion this is well-defined. Aim to show that and therefore that is the only possible root of .

- Assume . If is a root with , let . Then is a root with , contradicting the choice of .

- Assume . Choose a root with and let . Then is a root with , contradiction.

- Assume . Choose a root with . If is a root then is also a root. Therefore assume WLOG that for some in .

Then is a root. Similarly is a root, and , etc.

Continue in this way to show the existence of an infinite number of distinct roots of the polynomial equation , another contradiction.

Therefore is the only possibility, and consequently is the only possible root. Therefore write where and is a constant.

Use the functional equation to get , and so or . - Assume . If is a root with , let . Then is a root with , contradicting the choice of .