2.1 Definition

Definition 2..3  

1)

${\bf L}(I)$ denotes the ideal of $K[\underline{x}]$ generated by $\{L(f) \vert f \in I\}$.

2)

$f_1, \ldots, f_s \in I$ is called a standard basis of I if $\{L(f_1), \ldots, L(f_s)\}$ generates the ideal $L(I) \subset K[\underline{x}]$.

SINGULAR example: A standard basis computation:

// define a ring R= (Z/32003)[x,y,z]
ring R = 32003, (x,y,z), dp ;
// define 3 polynomials
poly s1  = x^3*y^2 + 151*x^5*y + 169*x^2*y4
          + 151*x^2*y*z3 + 186*x*y^6 + 169*y^9;
poly s2  = x^2*y^2*z^2 + 3*z^8;
poly s3  = 5*x^4*y^2 + 4*x*y^5 + 2x^2*y^2*z^3 + y^7 + 11*x^10;
// define the ideal i generated by s1,s2,s3
ideal i = s1, s2, s3;
// compute standard basis j of i
ideal j = std(i);
// display j;
j;