Singular

Dear Singular Team,

I work with a ring with coefficients of the rational numbers with several variables. The ordering is dp. I would like to eliminate some variables by using the [b]eliminate ( ideal_expression, product_of_ring_variables )[/b] command. Everything works well. I am interested in the different kinds of outputs if I define the ring with different orders of the ring variables.

It seems to me that the output is always independent of the internal order of those variables which occur in product_of_ring_variables. Please confirm or deny the validity of this.

Example: the output of the code below will not change if the variables "v7,v8,v9,v10,v11,v12,v13,v14,v15,v16,v19" will be changed to a different internal permutation (say "v19,v8,v9,v10,v11,v12,v13,v14,v15,v16,v7" or "v9,v8,v7,v11,v10,v12,v19,v14,v15,v16,v13"):
[code]
ring r=0,(
v3,v4,v5,v6,
v7,v8,v9,v10,v11,v12,v13,v14,v15,v16,v19),dp;
ideal i=
v7*v6+-1*v8*v5,
v8*v6+v7*v5+-1*v6*v4+-1*v5*v3,
-1*v9*v4+v10*v3,
-1*v10*v4+v6*v4+-1*v9*v3+v5*v3,
v11*v8+-1*v12*v7+-1*v11*v4+v7*v4+v12*v3+-1*v8*v3,
v11*v10+-1*v12*v9+-1*v11*v6+v9*v6+v12*v5+-1*v10*v5,
-1*v13*v6+v14*v5+v13*v4+-1*v5*v4+-1*v14*v3+v6*v3,
-1*v14*v6+-1*v13*v5+v14*v4+v13*v3,
v15*v14+-1*v16*v13,
v15*v10+-1*v16*v9+-1*v15*v6+v9*v6+v16*v5+-1*v10*v5,
-1+-1*v19*v15+v19*v11;
ideal e=eliminate(i,v19*v16*v15*v14*v13*v12*v11*v10*v9*v8*v7);
e;
[/code]
The outputs may differ only if the internal order of the variables not to eliminate are changed. E.g., instead "v3,v4,v5,v6" we may put "v4,v3,v6,v5", and this may give a different output.

Thank you for your answer in advance.