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shmoe
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#4
Aug24-06, 08:57 AM
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Something i didn't know about polynomials


Quote Quote by -Job-
Well, i know for x^k the last row of differences, before the zeros, is k!, but i haven't had much time to look into a generalized version.
What about x^k+x^l where l<k? How could you get the triangle for x^k+x^l given the triangles for x^l and x^k seperately?

Quote Quote by -Job-
It's also interesting to know that only the d+1 numbers on the left diagonal are required to produce all the numbers in the sequence (unlike the prims).
Have you tried to reconstruct your polynomial given this left diagonal?

Quote Quote by -Job-
I imagine that in computer science this can be used for compression. But this isn't surprising, i remember, when i was studying interpolation and numerical methods, noticing that any polynomial generated sequence can't be compressed in less than d+1 numbers.
You can't really do any better than just storing the coefficients. "Randomish" data isn't likely to fit on a polynomial of lower degree to make compression possible.

Quote Quote by -Job-
What about this left diagonal of d+1 numbers then? I imagine it can't possibly be polynomial itself.
You may know this, but given any sequence of numbers x(1),..,x(n) you can find a degree n-1 polynomial where f(i)=x(i). So when you are saying this sequence (or others) can't be a polynomial, what do you mean exactly? Some conditions on the degree? You will be able to say something about the minimum degree required from the difference table.