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