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Re: homogenous polynomials

  1. Jan 24, 2005 #1
    I'm having a problem with a proof I came across in one of my calculus books but it's not the calculus part of the proof that I'm having trouble with. Here's the actual proof:

    "Prove: The number of distinct derivatives of order n is the the same as the number of terms in a homogeneous polynomial in m variables of degree n"

    I've got a good idea about how to prove the part about the "number of distinct derivatives," so here, finally, is MY actual problem:

    Prove that the maximum number of terms possible in a homogeneous polynomial of m variables and degree n is given by

    [tex] \frac {(n + m -1) !} {n ! (m - 1) !}

    Let me know if it needs further explanation; I may not have done a good job explaining my problem.
  2. jcsd
  3. Jan 24, 2005 #2


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    Ah, counting, I like counting!

    You have n factors, each of which is one of m variables, and the order doesn't matter. (xxy and xyx aren't distinct)

    There are a couple of ways of modelling such problems. One way is as balls in boxes:

    You have n balls you want to distribute amonst m boxes. How many different arrangements are there?

    Though, I think what you want to do is to sort the factors, and partition them into m groups, so the first group corresponds to the first variable, the second group to the second variable, etc. Then, you can ask the question:

    I have n objects. How many ways can I place m-1 dividers into these objects, partitioning them into m distinct (possibly empty) groups?

    Actually, this problem is easier to solve if you can convert it into a similar problem where each group has at least one element...
  4. Jan 25, 2005 #3
    The balls in the boxes is an interesting (new to me) way to look at it. I'll give that a try and see if I can make any progress. Thanks.
  5. Feb 5, 2005 #4


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    this seems like a tautology to me. a derivative of degree n is determined, since the order of differentiation is unimportant (with a few hypotheses), by choosing which of the m variables to differentiate wrt, and a total of n of them. that is exactly what it means to choose a monomial of degree n in m variables.

    i.e. there is nothing to prove.
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