On 2016-03-30 16:08, Marc Mezzarobba wrote:
I think it is just a more or less arbitrary function that satisfies
quo*q + rem == p and extends the Euclidean division of univariate
polynomials. (But // can still be useful for exact divisions or
divisions by monomials...)

Is there any reason that quo_rem is not consistent with .mod()? That is no longer arbitrary and would also extend the Euclidean division of univariate polynomials.

Is multivariate division in the sense of
https://en.wikipedia.org/wiki/Gr%C3%B6bner_basis#Reduction implemented
somewhere in Sage?

Yes: p.mod(ideal) or ideal.reduce(p).

Thanks for the help. Note that p.mod(list of generators) also works.

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