  | ### 7.3.4 division (plural) `Syntax:` `division (` ideal_expression`,` ideal_expression `)` `division (` module_expression`,` module_expression `)` `division (` ideal_expression`,` ideal_expression`,` int_expression `)` `division (` module_expression`,` module_expression`,` int_expression `)` `division (` ideal_expression`,` ideal_expression`,` int_expression`,` intvec_expression `)` `division (` module_expression`,` module_expression`,` int_expression`,` intvec_expression `)` `Type:` list `Purpose:` `division` computes a left division with remainder. For two left ideals resp. modules `M` (first argument) and `N` (second argument), it returns a list `T,R,U` where `T` is a matrix, `R` is a left ideal resp. a module, and `U` is a diagonal matrix of units such that `transpose(U)*transpose(matrix(M))=transpose(T)*transpose(matrix(N)) + transpose(matrix(R))`. From this data one gets a left standard representation for the left normal form `R` of `M` with respect to a left Groebner basis of `N`. `division` uses different algorithms depending on whether `N` is represented by a Groebner basis. For a GR-algebra, the matrix `U` is the identity matrix. A matrix `T` as above is also computed by `lift`. For additional arguments `n` (third argument) and `w` (fourth argument), `division` returns a list `T,R` as above such that `transpose(matrix(M))=transpose(T)*transpose(matrix(N)) + transpose(matrix(R))` is a left standard representation for the left normal form `R` of `M` with respect to `N` up to weighted degree `n` with respect to the weight vector `w`. The weighted degree of `T` and `R` respect to `w` is at most `n`. If the weight vector `w` is not given, `division` uses the standard weight vector `w=1,...,1`. `Example:`
      LIB "dmod.lib";
      ring r = 0,(x,y),dp;
      poly f = x^3+xy;
      def S = Sannfs(f); setring S; // compute the annihilator of f^s
      LD; // is not a Groebner basis yet!
      ==> LD[1]=3*x^2*Dy-x*Dx+y*Dy
      ==> LD[2]=x*Dx+2*y*Dy-3*s
      poly f = imap(r,f);
      poly P = f*Dx-s*diff(f,x);
      division(P,LD); // so P is in the ideal via the cofactors in _[1]
      ==> [1]:
      ==>    _[1,1]=-2/3*y
      ==>    _[2,1]=x^2+1/3*y
      ==> [2]:
      ==>    _[1]=0
      ==> [3]:
      ==>    _[1,1]=1
      ideal I = LD, f; // consider a bigger ideal
      list L = division(s^2, I); // the normal form is -2s-1
      L;
      ==> [1]:
      ==>    _[1,1]=2/3*x^2*Dy-1/3*x*Dx+2/3*s+1/3
      ==>    _[2,1]=2/3*x^2*Dy-1/3*x*Dx-1/3*s-2/3
      ==>    _[3,1]=-2*x*Dy^2+Dx*Dy
      ==> [2]:
      ==>    _[1]=-2*s-1
      ==> [3]:
      ==>    _[1,1]=1
      // now we show that the formula above holds
      matrix M[1][1] = s^2; matrix N = matrix(I);
      matrix T = matrix(L[1]); matrix R = matrix(L[2]); matrix U  = matrix(L[3]);
      // the formula must return zero:
      transpose(U)*transpose(M) - transpose(T)*transpose(N) - transpose(R);
      ==> _[1,1]=0
See ideal; lift; module; poly; vector.

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