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If I have a multivariate polynomial $P[X_1,\dots,X_n]\in \mathbb{R}[X_1,\dots, X_n]$, is there a polynomial time algorithm to factor the polynomial into irreducible polynomials $\in \mathbb{C}[X_1,\dots,X_n]$?

Also, how do the factors look like - are they all linear or constant polynomials or could they also be non-linear?

If the factorization involves non-linear irreducible polynomials, then is it possible to list all the roots of $P[X_1,\dots,X_n]$ in polynomial time?

Edit: To give some motivation for my question, my goal is to find if a polynomial is $\geq 0$ on its entire domain or not.

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    For th$e$ edited question, there are many algorithms, but probably none that involve factoring.2011-10-14

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