@MISC{POLYNOM1.MIZ,
  AUTHOR = {Rudnicki, Piotr and Trybulec, Andrzej},
  TITLE = {Multivariate Polynomials 
   with Arbitrary Number of Variables},
  NOTE = {This work has been supported by NSERC Grant OGP9207 and
NATO CRG 951368.},
  SECTION1 = {Basics},
  SECTION2 = {Sequence Flattening},
  SECTION3 = {Functions Yielding Natural Numbers},
  SECTION4 = {The Support of a Function},
  SECTION5 = {Bags},
  SECTION6 = {Formal Power Series},
  SECTION7 = {Polynomials},
  SECTION8 = {The Ring of Polynomials},
  DAY = {22},
  MONTH = {September},
  YEAR = {1999},
  ADDRESS1 = {University of Alberta\\Edmonton},
  ADDRESS2 = {University of Bia{\l}ystok},
  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 \cite{DTCONSTR.ABS}, 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.}}

