| #### D.4.34.1 markov4ti2 Procedure from library `sing4ti2.lib` (see sing4ti2_lib). Usage: markov4ti2(A[,i]); A=intmat i=int Assume: - A is a matrix with integer entries which describes the lattice as ker(A), if second argument is not present, as left image Im(A) = {zA, z \in ZZ^k}(!), if second argument is a positive integer - number of variables of basering equals number of columns of A (for ker(A)) resp. of rows of A (for Im(A)) Create: files sing4ti2.mat, sing4ti2.lat, sing4ti2.mar in the current directory (I/O files for communication with 4ti2) Note: input rules for 4ti2 also apply to input to this procedure hence ker(A)={x|Ax=0} and Im(A)={xA} Return: toric ideal specified by Markov basis thereof Example:
    LIB "sing4ti2.lib";
    ring r=0,(x,y,z),dp;
    matrix M[2][3]=0,1,2,2,1,0;
    markov4ti2(M);
    ==> _[1]=-y2+xz
    matrix N[1][3]=1,2,1;
    markov4ti2(N,1);
    ==> _[1]=xy2z-1
    ==> _[2]=xy2z-1

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