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Some Properties of Enumeration in the Theory of Modular Systems
Author(s) -
MacAulay F. S.
Publication year - 1927
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s2-26.1.531
Subject(s) - enumeration , modular design , citation , computer science , mathematics , information retrieval , algebra over a field , combinatorics , library science , programming language , pure mathematics
The object of this note is to discover the limiting relations which must •exist between the terms of the series Do, Dv ..., A, ••• where A is the number of linearly independent homogeneous polynomials of degree I (or of degrees less than or equal to I in the case of non-homogeneous polynomials) belonging to some actual modular system, the system being either perfectly general or else general of its kind. A modular system (or in modern nomenclature a polynomial-ideal) is denned to be any aggregate of polynomials in n variables such that the sum of any two, and also the product of any one by a constant* or any of the variables xlf x.2, ..., x.u, belongs to the ideal •or aggregate. The converse and more important question is that of finding the actual values of Do, Dif D.2) ... for a given polynomial-ideal, that is, an ideal denned by the stated conditions which its members individually and collectively have to satisfy. But with few exceptions, some of which will be noted later, no general answer can be found to this converse question. In a classical memoir, Hilbertt has shown that when I is large enough Di becomes a polynomial in I; and OstrowskiJ has employed Hilbert's process