David R. Stoutemyer
Abstract
Polynomials over the Gaussian integers Z[i] are a unique factorization domain, and the units in Z[i] are 1, -1, i and --i. To implement the advantageous uniqueness, we might have to factor one of these units out of some multinomial factors to make all of the factors unit normal. Not doing so can lead to unnecessarily complicated results, disastrous unrecognized division by 0, and lost opportunities to obtain closed-form integrals, sums, limits, equation solutions, etc. For example, the default simplification of at least two major computer algebra systems don't recognize that the linear factors in the numerator and denominator of the following ratio are proportional, hence do not simplify the ratio to 1: --i ((1 + i) z + i) / (1 -- i) z + 1 →1.
Unlike polynomials over the integers or rational numbers, the literature doesn't appear to address unit normalization over the Gaussian integers. Therefore this note proposes two alternative definitions. They are easy to implement, execute quickly, and can't greatly increase the bulk of an expression. Consequently they are good candidates for inclusion in the default simplification of every computer algebra system.
- D.R. Stoutemyer, Ten commandments for good default expression simplification, Proceedings of the Milestones in Computer Algebra Conference, 2008, http://www.orcca.on.ca/conferences/mica2008/Google Scholar
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