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.
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:
where special means: gcd(p,Dp)= p.
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:
Which is easy to understand.
In other cases, the definitions are used quite different. For Instance:
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.
The example I give above:
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
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].