Is the Polar Decomposition Finitely Computable?

The polar decomposition of a square matrix is a major step toward the singular value decomposition, and is an important device in its own right. Singular values of a matrix $A$, being the eigenvalues for a matrix closely related to $A$, generally cannot be computed by a finite process if only arithmetic operations and radicals are allowed. However, this consideration alone does not prove that the polar decomposition is not finitely computable. The problem of finite computability of the polar decomposition is not settled in this paper, but we do show it to be equivalent to the following simpler-looking problem. Suppose $f$ is a real polynomial of degree $ n > 4$, and all the roots of $f$ are distinct positive numbers. Denote by $g$ a polynomial of the same degree whose zeros are the positive square roots of the zeros of $f$. Can this polynomial $g$ always be computed finitely for a given polynomial $f$? In the Appendix we discuss one nontrivial situation where the polar decomposition can indeed be computed finitely.