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Math Help - Problems with abstract algebra (differential fields) in algorithms

  1. #1
    ASL
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    Problems with abstract algebra (differential fields) in algorithms

    Hi,

    I am an engineer working with some algorithms that uses a lot of abstract algebra. And as I explain below, this is sometimes a bit difficult for me to understand.

    The abstract algebra involved are: differential fields, and field extensions.

    The extensions fields are simple transcendental differential extensions to the field k. If K denotes such an extension field, t is said to be in K and a monomial over k, and Dt is a polynomial in k[t] -- where D is the derivation in the field.

    The book, that I am reading defines some sets of polynomials:

    <br />
S_{k[t]:k} = \{ p \in k[t] \text{ such  that p is special w.r.t D}\}<br />

    <br />
S_{k[t]:k}^{irr} = \{ p \in S_{k[t]:k} \text{ such  that p is monic and  irreducible}\}<br />

    where special means: gcd(p,Dp)= p.

    And

    <br />
S_{1,k[t]:k} = \{ p \in k[t] \text{ such  that p is special of the first kind}\}<br />

    <br />
S_{1,k[t]:k}^{irr} = \{ p \in S_{1,k[t]:k} \text{ such  that p is monic and  irreducible}\}<br />

    Note, that it is necessary to know the precise meaning of "special of first kind". It is just some advanced math related to a polynomial in k[t]. Note also, that when the monomial extension is clear from the context, the subscripts are omitted.

    My problem with these definitions are, that I do not always understand how to interprete them, when they occur in the pseudocode for the various algorithms listed in the book.

    Often the following definition is used:

    <br />
p \in S^{irr}<br />

    Which is easy to understand.

    In other cases, the definitions are used quite different. For Instance:

    <br />
\text{if }  S_1^{irr} = S^{irr}  \text{ then ...}<br />

    This, I find difficult to understand, since I cannot figure out how to relate the definitions (sets) to a specific polynomial, or other elements in the context.

    If the two sets are equal, what makes them equal?, and how are their content defined?

    Hope someone can help me a bit on this.

    Kind regards,

    ASL
    Last edited by ASL; August 13th 2010 at 10:25 AM.
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  2. #2
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    Maybe the choice of extension field with differential can make the two sets equal? It would be easier to help if you gave more context.
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  3. #3
    ASL
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    Re: Problems with abstract algebra (differential fields) in algorithms

    The example I give above:

    <br />
\text{if }  S_1^{irr} = S^{irr}  \text{ then ...}<br />

    occurs in an algorithm that takes as input: the derivation D, and some functions f, and w1,w2,.... in k(t).

    The text explaining the algorithm writes: this gives an algorithm that reduces the problem to one over k(t) if S_1^{irr} = S^{irr}

    And you may be right.

    It is probably the monomial t in the extension field which makes the sets equal. Or more precise: Dt which is a polynomial in k[t].
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