Suppose we are given a representation of a finite series of natural numbers:

$\sum_{i=0}^N{c_i x^i}$

The representation is essentially an expression that is a rational function of two polynomials.

Is it possible to add/subtract this series repeatedly to get a result that contains only part of the series?

A simple formulation of the problem

We can forget that we have a series, for a moment, and consider it more like a tile. It contains a set of numbers on it that are in a particular order that cannot be changed. It's basically striaght; it just contains this set of numbers on a line.

Now we actually have many copies of this same tile, and some are oriented differently. The different orientation comes from the fact that the tile could be on a line into another dimension. For example, one tile may start at coordinates (0,0) and end at coordinates $(0,N)$. Another may start at (0,0) and end at $(N,0)$. All tiles are essentially the same ordered set of numbers that start at one point in $m$ dimensions and end at another point.

There is essentially one operation that we can perform. We can take the difference of numbers at a given point. For example, if the sequence is (1,2,3), we can take one tile that starts at (0,0) and proceeds to (0,2). Subtract another tile that starts at (0,0) and proceeds to (2,0). This would cancel out the ones in both tiles, since we subtract the one at (0,0) in the second tile from the one at (0,0) in the first tile.

I'm wondering if we can somehow add and subtract these tiles so that only a single number remains. It may be that more than one location contains this same number.

There are some rules, so an example and explanations will probably help.

Could you give an example?

Here's an example. Consider the finite series $10 + 200x$.

We can repeatedly add and subtract (scalar multiples of) this series (and a similar series) in the $x,y$ "plane" to end up with $200x$, only part of the series.

Here's how:

(0) Start with $10+200x$.

(1) Subtract $10 + 200y$. This eliminates 10 and we're left with $200x-200y$.

(2) Add $(10+200x)\cdot y / x$. This eliminates the $-200y$ in the previous result and adds $10\cdot y / x$, resulting in $200x+10\cdot y / x$.

(3) Subtract$(10+200y)\cdot y / x$. This eliminates $10\cdot y / x$, subtracts $200\cdot y^2 / x$, and we're left with $200x - 200\cdot y^2 / x$

(4) Add $((10+200x)\cdot y / x)\cdot y / x$. This eliminates the $-200\cdot y^2 / x$ in the previous result and adds $(10\cdot y / x)\cdot y / x$, resulting in $((200x+10)\cdot y / x)\cdot y / x$.

...

We repeatedly add and subtract pieces of the series this way. It can be shown that the result of this infinite series is $200x$, which is only part of the original series.

I'd like to know if we can add and subtract larger series similarly (but possibly in more dimensions) to end up with only a single coefficient or piece of the series.

For example, If we consider the series $1 + 2x + 3x^2$, we may be able to eliminate $1$ and $3x^2$ from this series by a set of careful additions and subtractions.

I don't know which branches of mathematics this deals with, but I'm hoping an expert can provide me with some direction.

Why are you doing this/ What's your motivation?

Results from this "puzzle" would help speed up an algorithm significantly, so I'm interested in publishing this result with whoever helps me. I know MO's position on algorithms, but I believe that this is more of a "tiling" or mathematics question than a question on an algorithm.

I'd like to know all branches of mathematics that deal with this question, and where I can go, or who I can go to, that will help me solve the general problem.

Some additional notes: The value of $x$ is not allowed to be modified. We can rewrite $x$ as another variable (I rewrote $x$ as $y$ above, for example). We are also allowed to "shift" the series (for example, multiply by $y / x$ as done above). Cancellation is, of course, allowed - and it may be done infinitely many times. These are the only allowable operations.

Also, I am particularly interested in the coefficient in the middle of the series.

One additional consideration: methods that use fewer variables are preferred.