Quotient decomposition by Groebner basis

asked 2011-01-08 21:54:12 +0100

mhampton
399 ●6 ●10 ●19

I can accomplish the following task in awkward ways using syzygy modules, but I am wondering if there is a better way somehow. It would be nice to have a single command for it.

Suppose we have a polynomial $P$ and a set of polynomials $Q_1,...,Q_n$, and it is possible to calculate the Groebner basis $G$ of the ideal generated by all the $Q_i$. Let $R$ be the remainder of $P$ after reducing by $G$. In Sage, how can we find polynomials $S_1,...,S_n$ such that $P = R + \sum S_i Q_i$?

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answered 2011-01-13 11:27:31 +0100

malb
71 ●2 ●4

Do you mean this?

sage: R.<x,y,z> = GF(127)[]
sage: p = R.random_element(degree=3)
sage: Q = [R.random_element() for _ in range(R.ngens())]
sage: r = p.reduce(Ideal(Q))
sage: M = (p - r).lift(Q)
sage: p -r == sum([M[i]*Q[i] for i in range(len(M))])
True