scratch
§ Why Division Algorithm with Multiple Variables Go Bad
created 2021-10-04
- In
C[x, y], defining division is complicated, and needs grobner bases to work. - It's because they don't obey the GCD property. Just because
gcd(a, b) = g does not mean that there exist k, l such that ak + bl = g - For example, in
C[x, y], we have gcd(x, y) = 1 but we don't have polynomials k, l such that kx + ly = 1. - Proof: suppose for contradiction that there do exist
k, l such that kx + ly = 1. Modulo x, this means that ly = 1 which is absurd, and similarly modulo x it means kx = 1 which is also absurd.
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