I know that in order to factor a one dimensional polynomial one can find the roots with some method, for instance a numerical newton method. Then one can systematically divide with $(variable-root)$ for each root found and then be done. Is there any analogous way to do this for multivariate polynomials? Does there exist any "unique" or "natural" factorization for those? It is obvious we can do this in the case our polynomial is separable i.e. $$P(x_1,x_2,\cdots,x_n) = P_1(x_1)P_2(x_2) \cdots P_n(x_n)$$ because then we could just factor each $P_k(x_k)$ separately.
But what about the general case?
Best Answer
The factorization of multivariate polynomials is quite a difficult topic. In brief, the polynomial is converted into a univariate one (one in one variable), by selecting suitable values for the other variables.
The resulting univariate polynomial is then factorized - there are several methods to do this in the general case. The factored polynomial is then 'built up' by using something called Hensel lifting. This is quite an advanced subject area and would typically be taught sort of on 3rd year abstract algebra university courses.
The multivariate factorization problems you get in school or whatever tend to be fairly simple 'toy problems' that have nice neat solutions, by usually employing something like difference of two cubes, or by exploiting some sort of substitution.