h = quotient(f, g)
h = f // g
This function produces a matrix h such that f - g*h is the reduction of f modulo a Gröbner basis for the image of g. See quotient' for the dual notion.
If the remainder f - g*h is zero, then the quotient f//g satisfies the equation f === g * (f//g). Otherwise, the equation g * h + r === f will hold, where r is the map provided by remainder.
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Note, however, that currently the function remainder only works if the source of f is free.
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f // g
If either f or g is a ring element, then it is taken to be a scalar matrix acting on H. If g is a MonomialIdeal, then it is taken to be the matrix of generators of g. Finally, if g is a GroebnerBasis object, then the Gröbner basis as so far computed is used. In these latter two cases, no Gröbner bases will be computed.
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One common use is the following. If an ideal contains 1, then we may write 1 in terms of the generators of the ideal. First we make an ideal.
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Transposing the (row) matrix of generators of the ideal puts the generators on separate lines and shows the degrees.
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Next we test whether 1 is in the ideal.
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We see that 1 is in the ideal. Now we represent 1 in terms of the generators of I.
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f // g
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The source of this document is in Macaulay2Doc/functions/quotient-remainder-doc.m2:126:0.