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Solutions of Polynomial Equations And Ideals

Let tex2html_wrap_inline4136 denote the set of (noncommutative) polynomials in the indeterminates
tex2html_wrap_inline4138 over a field K. tex2html_wrap_inline4136 is a noncommutative algebra. Let tex2html_wrap_inline4144 tex2html_wrap_inline4136 , C be a set of polynomials tex2html_wrap_inline4150 and tex2html_wrap_inline4152 with each tex2html_wrap_inline4154 . If r is a common zero of tex2html_wrap_inline4158 , then r is a zero of every polynomial p which lies in the smallest ideal tex2html_wrap_inline4164 generated by C. Furthermore, if tex2html_wrap_inline4168 is a generating set for the ideal tex2html_wrap_inline4164 , 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 tex2html_wrap_inline4178 lie in a commutative algebra and they also hold when the tex2html_wrap_inline4178 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 tex2html_wrap_inline4158 directly, it could be advantageous to find a different set tex2html_wrap_inline4184 of polynomials which generates the same ideal as tex2html_wrap_inline4150 and then solve for the common zeros on the q's. For example, tex2html_wrap_inline4190 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 §.

Wed Jul 3 10:27:42 PDT 1996