Manifolds of difference polynomials
Manifolds of difference polynomials
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DOI：
10.1090/s0002-9947-1948-0025438-x
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发表时间：
1948
影响因子：
1.3
通讯作者：
R. Cohn
中科院分区：
文献类型：
--
作者：
R. Cohn
1.这是本文的目的是发展在一些详细的结构的流形所确定的系统的差异多项式。我们的结果将必然局限于抽象域上的多项式的情况下，因为一个合适的解析差分方程的存在定理是不可用的。理想理论是由J.F. Ritt和H. W. Raudenbush（1）的差分多项式的抽象系统，因此，在我们的工作中是基本的。2.在本文的第一部分中，我们描述了一种消除代数差分方程组中未知数的理论方法。我们利用这种方法证明了域扩张上代数基本定理的差域的类似物。借助于这些结果，我们在定理III中证明了在素差理想中任意（2）个未知数的个数对于任意未知数集合的所有可能选择都是常数。3.第二部分是关于抽象域中单个代数不可约差分多项式的流形。解析域中多项式的因式分解是由J. F. Ritt（3）在一阶差分多项式流形的分解中，确定不为零阶多项式所保持的不可约流形的最大个数。在定理IV中，我们证明了，当Ritt分解过程应用于抽象域中的多项式A时，它产生的每个多项式序列实际上确定由A持有的素理想，但不是由任何比A低阶的多项式持有的素理想。此外，所有这样的素理想都是这样得到的。这构成了抽象域上差分多项式存在定理的一种形式，并且是理论进一步发展的基础。由因子分解过程确定的A的不可约流形称为A的通解。我们将看到，如果A是一阶的，则所有解都包含在通解中。这一结果证实，在一个基因-
1. It is the purpose of this paper to develop in some detail the structure of the manifolds determined by systems of difference polynomials. Our results will necessarily be confined to the case of polynomials in an abstract field, since a suitable existence theorem for analytic difference equations is not available. The ideal theory, developed by J. F. Ritt and H. W. Raudenbush(1) for abstract systems of difference polynomials, is therefore fundamental in our work. 2. In Part I of our paper we describe a theoretical method for elimination of unknowns in systems of algebraic difference equations. We employ this method to prove analogues for difference fields of fundamental theorems of algebra on field extensions. With the aid of these results we show in Theorem III that the number of arbitrary(2) unknowns in a prime difference ideal is constant for all possible choices of sets of arbitrary unknowns. 3. Part II is concerned with the manifold of a single algebraically irreducible difference polynomial in an abstract field. A factorization process for polynomials in analytic fields was developed by J. F. Ritt(3) in determinining the maximum number of irreducible manifolds, not held by polynomials of zero order, in the decomposition of the manifold of a first order difference polynomial. In Theorem IV we show that, when the Ritt factorization process is applied to a polynomial A in an abstract field, each of the polynomial sequences it produces actually determines a prime ideal held by A but not by any polynomial of lower order than A. Furthermore, all such prime ideals are obtained in this way. This constitutes a form of existence theorem for difference polynomials in abstract fields, and is fundamental in the further development of the theory. The irreducible manifolds of A determined by the factorization process we call the general solution of A. We shall see that, if A is of first order, all solutions are included in the general solution. This result confirms, in a gen-