Why is studying polynomials important
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`` De todos es conocido que los polinomios constituyen el núcleo del álgebra '' (Jaime Gutierrez)Polynomials and polynomial functions have always enjoyed great popularity. They owe their popularity to the fact that polynomial functions can be represented by a (finite) term. Such a term also indicates how the function is built up from basic functions (in the simplest (= univariate) case from the constant functions and the identity). The term representation of a function should not only be seen as a data compression, that is, as a possibility to represent functions on possibly infinite sets using finite terms, but also as a clear and meaningful way of specifying a function. The concept of the polynomial function can be generalized quite easily from rings to universal algebras. All those functions are then called polynomial functions that can be built from constant functions and the identity with the help of the operations available in algebra. It can happen that two functions built up from the basic functions in different ways turn out to be the same, that is, that different terms induce the same function. For example, the terms induce and same function on the rational numbers. In our case this follows from the (right) distributive law. We will study methods of early detection of such similarities. A polynomial will then be an equivalence class of terms which lead to the same function due to equations such as the associative or distributive law. That will be the content of the first chapter.
In the second chapter we will deal with whether the early detection of such equality is also effective, i.e. algorithmically, possible. We first show for important special cases (i.e. for groups, Abelian groups, rings) what the polynomials look like over these algebras. Then we mention a method of how one can algorithmically treat the problem of `` equality modulo given identities '' (such as the distributive law). The appendix contains simple experiments with an exciting method, a completion algorithm that uses the idea of critical pairs and was developed by D. Knuth and P.Bendix in 1967.
Then we turn away from the polynomials and devote ourselves again fully to the polynomial functions. These allow an important characterization of the ideals of -Groups (these are groups with further operations, e.g. multiplications).
Unfortunately, not every function in life is a polynomial function. Therefore we will try to interpolate as many functions as possible using polynomial functions. The same operations can be carried out on the polynomial functions by applying them point by point as on the underlying algebras. However, this does not fully do justice to the nature of functional algebras, since in this way we treat functional algebras as if they were merely direct products of the underlying algebras (or sub-algebras of the direct product). In order to be able to consider the functions not only point by point, we create their execution one after the other. A set of functions on a group that is completed under pointwise addition and under the sequential execution is called a (function) fast ring. We lead the fast ring of n Set by a (small, clear) function fastring (such as the polynomial functions) interpolable functions ( ), which will allow us to present many results from interpolation theory in a clear manner. A typical result from this atmosphere is the following sentence:
If you can already interpolate at four points on a group with a fast ring of functions, then also at any (but finite) number of points.We know that the polynomial functions on the real numbers have the property of interpolation, that is, we can always go through n Points in the plane with different x-Coordinates lay the graph of a polynomial function. The above theorem shows us that it is sufficient to prove this result for four points and, when proving this real interpolation property, relieves us of calculating the Vandermonde determinant for n > 4 that we usually use to help show this theorem. Furthermore, we characterize all groups, rings and fast rings on which every function is a polynomial function or on which at least every function can be interpolated at any but finite number of places by a polynomial function. Such algebras are called polynomial-complete or locally polynomial-complete. The polynomial completeness of algebras has so far been successfully investigated using two methods in particular: The interpolation theory, as practiced, for example, by H.K. Kaiser and A.F. Pixley and (for -Groups) with the density theorems of the Fastring theory, which go back to H.Wielandt and G.Betsch. In the fourth chapter we prove the density theorem for zero-symmetric, 2-primitive fast rings with 1 and the interpolation theorem for zero-symmetric, 0-primitive fast rings with the aid of a superordinate interpolation theorem for fast rings, the proof of which uses the ideas of interpolation theory. This superordinate interpolation theorem helps us to get the known results about polynomial-complete -Groups that we already know as a consequence of the Fastring density theorems. I would also like to present the ideas with which R.Mlitz succeeded in formulating a density theorem for universal algebras, which implies both the density theorem for rings and that for fast rings.
We will also see that there are limits to interpolation using polynomial functions: There are essentially two obstacles to polynomial completeness: Sometimes all polynomial functions are of the form `` linear function + constant function ''. The deeper reason that in this case the polynomials are often not sufficient to interpolate arbitrary, i.e. also non-linear functions, is that the fast ring of the polynomial functions then satisfies equations (such as linearity) that do not apply in the fast ring of the functions to be interpolated . If all polynomial functions are `` linear + constant '', at least the functions of the form `` linear + constant '' can be interpolated sufficiently well. To show this, one uses Jacobson's density theorem from ring theory. Polynomials have the second property that is detrimental to their interpolation ability, namely respecting congruence relations on algebras. So at best we can interpolate the functions that also respect the congruence relations. Such functions are called compatible. Algebras in which every compatible function is already a polynomial function are called affine complete. We prove a version of the interpolation theorem for fast rings that sometimes guarantees affine completeness. Then we shall briefly consider how the sentences about single-digit functions can be extended to sentences about multi-digit functions.
In the fifth chapter we deal with a specific problem: Be n a given natural number. What functions on the whole numbers can at n arbitrarily selected points are interpolated by a polynomial function (which is induced by a polynomial with integer coefficients)? We get a full characterization of these functions. Or, to put it in the language of G. Pilz's Fastringbuch, we determine explicit.
I would like to thank the supervisor of this diploma thesis, Prof. Dr Günter Pilz, for his support and support. I owe the countless discussions with him about mathematics not only the creation of this thesis, but also the drive and self-confidence that went for doing math are irreplaceable.
But my most important sponsors were my parents and my brothers, who tried to convey joy in thinking to me from early childhood.
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Tue Jan 7 15:27:47 MET 1997
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