Take the 2-minute tour ×  I had hoped that this would be easier.
I have power series that I'm using. I want to know what simple
structure describes the operations that I'm performing on them.
It would be better if I could find something that non-
mathematicians (actually computer scientists) would understand,
but I'd like to know the technical terminology too.  First, let
me define three power series, $f(x)$, $g(x)$, and $h(x)$:  $f(x)
= \sum_{i=m_1}^{n_1}{c_i x^i}$  $g(x) = \sum_{i=m_1}^{n_1}{d_i
x^i}$  $h(y) = \sum_{i=m_2}^{n_2}{e_i y^i}$  Then the two
operations that I perform are as follows:  (1) Subtraction $r(x)
= f(x) - g(x)$  $r(x) = \sum_{i=m_1}^{n_1}{c_i - d_i}$  (2)
Multiplication $r(x,y) = f(x)h(y)$  $r(x,y) =
\sum_{i=m_1}^{n_1}{\sum_{j=m_2}^{n_2}{c_i \cdot e_j}}$  All of
the coefficients and variables are integers.  I'm writing a
paper, and I'm searching for a more "definite" description than
what I have. It is somewhat ackward to describe, and I want to
convey this information exactly and concis