- Usage:
`(M,C) = coefficients f`

- Inputs:
`f`, a one-row Matrix with`n`columns, say, or a RingElement, to be interpreted as a one-by-one matrix. (A future implementation will handle matrices with more than one row.)

- Optional inputs:
`Variables =>`a list, default value null, a list`v`of variables. If a value for this option is not specified, all of the (top-level) variables are used.`Monomials =>`default value null, a list or one-row matrix of monomials, each of which is formed using just variables in`v`.

- Outputs:
`M`, a matrix, either the value of the`Monomials`option, if specified (converted to a one-row matrix, if necessary), or a one-row matrix of those monomials appearing in`f`that involve just variables in`v`, in descending order. Let`m`denote the number of columns it has.`C`, a matrix, the`m`by`n`matrix`C`such that`C_(i,j)`is the coefficient in`f_(0,j)`of the monomial`M_(0,i)`. In other words,`C`is the unique matrix not involving the (specified) variables such that`M*C == f`, unless a value was specified for the`Monomials`option that did not include all the monomials in the variables`v`used by`f`

i1 : R = QQ[a,b,c,d,e,f][x,y]; |

i2 : F = a*x^2+b*x*y+c*y^2 2 2 o2 = a*x + b*x*y + c*y o2 : R |

i3 : (M,C) = coefficients F o3 = (| x2 xy y2 |, {2, 0} | a |) {2, 0} | b | {2, 0} | c | o3 : Sequence |

i4 : M*C === matrix F o4 = true |

i5 : G = d*x^2+e*x*y+f*y^2 2 2 o5 = d*x + e*x*y + f*y o5 : R |

i6 : P = matrix{{x*F,y*F,x*G,y*G}} o6 = | ax3+bx2y+cxy2 ax2y+bxy2+cy3 dx3+ex2y+fxy2 dx2y+exy2+fy3 | 1 4 o6 : Matrix R <--- R |

i7 : (M,C) = coefficients P o7 = (| x3 x2y xy2 y3 |, {3, 0} | a 0 d 0 |) {3, 0} | b a e d | {3, 0} | c b f e | {3, 0} | 0 c 0 f | o7 : Sequence |

i8 : M*C === P o8 = true |

i9 : (M,C) = coefficients(P, Monomials=>{x^3,y^3,x^2*y,x*y^2}) o9 = (| x3 y3 x2y xy2 |, {3, 0} | a 0 d 0 |) {3, 0} | 0 c 0 f | {3, 0} | b a e d | {3, 0} | c b f e | o9 : Sequence |

i10 : (M,C) = coefficients(P, Monomials=>{x^3,y^3}) o10 = (| x3 y3 |, {3, 0} | a 0 d 0 |) {3, 0} | 0 c 0 f | o10 : Sequence |

i11 : M*C == P o11 = false |

- monomials -- matrix of monomials in a ring element or matrix

- coefficients(Matrix)
- coefficients(RingElement)