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### Solutions of Polynomial Equations And Ideals

Let denote the set of (noncommutative) polynomials in the indeterminates
over a field K. is a noncommutative algebra. Let , C be a set of polynomials and with each . If r is a common zero of , then r is a zero of every polynomial p which lies in the smallest ideal generated by C. Furthermore, if is a generating set for the ideal , then r is a common zero of the p's if and only if it is a common zero of the q's. These ideas are learned by most mathematics undergraduates in the case that the indeterminates lie in a commutative algebra and they also hold when the lie in a noncommutative algebra.

The practical value of doing this algebraic manipulation is that instead of trying to solve for the common zeros of directly, it could be advantageous to find a different set of polynomials which generates the same ideal as and then solve for the common zeros on the q's. For example, might consist of decoupled polynomial equations which could be solved numerically (say using Matlab) while the original set C produces numerically intractable problems.

The reader who is interested in seeing examples at this point might decide to skip directly to §.

Helton
Wed Jul 3 10:27:42 PDT 1996