# Factoring Systems of Polynomials

I was thinking sometime about solving multidimensional systems of polynomials in a way analgous to how polynomials are solved today that is:

Given $p(x) = a_0 + a_1x + a_2x^2 .. a_Nx^N$

1. Use Newton's Method or some faster method to approximate root to desired accuracy, called $r_i$ (ex: $r_0$, $r_1$ ... $r_N$)
2. Divide out $x - r_i$ from $p(x)$ to create $p_S(x)$ (ex: $p_1(x)$ $p_2(x)$ ... $p_N(x)$ = no factors)
3. Truncate remainder and repeat process until no further divisions can be made

I was at a loss on how to carry out the same sort of algorithm for systems in higher dimensions until it occurred to me that what needed to be divided out weren't just singletons (like single expressions) but rather it was entire systems that needed to be divided out.

We divide out $x - r_i$ since it carries all the "information" about the root $r_i$ and hence we can remove it.

For higher dimensional systems (ex: 3d) we need to divide out entire systems as it takes a system of equations to specify a point in higher dimensions.

Is this reasonable intuition for trying to make an algorithm to do the job? Has this been done before?

My question was that there can be an infinite number of linear systems that have solutions at a particular point. What am I missing here?

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