> If the divisor is monic, then everything is okay, but if the divisor
> is not monic, it's not clear what the remainder should be. I took the
> agnostic option for the moment.

Why not make it agree with Magma's multivariate definition (used in  
their Euclidean ring Groebner basis calculations)?

The reference is

http://magma.maths.usyd.edu.au/magma/htmlhelp/text1115.htm#11136

I can elaborate.  For what it's worth, I have implemented some of  
this in a .sage file but I couldn't make it very general -- the  
polynomial hierarchy is hard to comprehend these days.

Nick

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