Journal of Formalized Mathematics
Volume 11, 1999
University of Bialystok
Copyright (c) 1999 Association of Mizar Users

## Multivariate Polynomials with Arbitrary Number of Variables

Piotr Rudnicki
University of Alberta, Edmonton
Andrzej Trybulec
University of Bialystok

### Summary.

The goal of this article is to define multivariate polynomials in arbitrary number of indeterminates and then to prove that they constitute a ring (over appropriate structure of coefficients).\par The introductory section includes quite a number of auxiliary lemmas related to many different parts of the MML. The second section characterizes the sequence flattening operation, introduced in [9], but so far lacking theorems about its fundamental properties.\par We first define formal power series in arbitrary number of variables. The auxiliary concept on which the construction of formal power series is based is the notion of a bag. A bag of a set \$X\$ is a natural function on \$X\$ which is zero almost everywhere. The elements of \$X\$ play the role of formal variables and a bag gives their exponents thus forming a power product. Series are defined for an ordered set of variables (we use ordinal numbers). A series in \$o\$ variables over a structure \$S\$ is a function assigning an element of the carrier of \$S\$ (coefficient) to each bag of \$o\$.\par We define the operations of addition, complement and multiplication for formal power series and prove their properties which depend on assumed properties of the structure from which the coefficients are taken. (We would like to note that proving associativity of multiplication turned out to be technically complicated.)\par Polynomial is defined as a formal power series with finite number of non zero coefficients. In conclusion, the ring of polynomials is defined.

This work has been supported by NSERC Grant OGP9207 and NATO CRG 951368.

#### MML Identifier: POLYNOM1

The terminology and notation used in this paper have been introduced in the following articles [33] [17] [43] [36] [44] [45] [46] [14] [20] [37] [32] [3] [35] [16] [15] [12] [13] [19] [2] [11] [39] [38] [42] [8] [18] [4] [24] [1] [5] [41] [27] [40] [28] [7] [22] [6] [23] [31] [10] [34] [9] [30] [29] [26] [25] [21] [47]

#### Contents (PDF format)

1. Basics
2. Sequence Flattening
3. Functions Yielding Natural Numbers
4. The Support of a Function
5. Bags
6. Formal Power Series
7. Polynomials
8. The Ring of Polynomials

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