// MATH 531: Intro to Singular /////////////////////////////////////////// 

                     SINGULAR                             /
 A Computer Algebra System for Polynomial Computations   /   version 2-0-4
                                                       0<
     by: G.-M. Greuel, G. Pfister, H. Schoenemann        \   October 2002
FB Mathematik der Universitaet, D-67653 Kaiserslautern    \
> 23+435;
458
> int k = 12;
> k;
12
> int j;
> j = 1 + k;
> _;
12
> 
. 
. ring r = 0, (x,y,z), dp;
> 
. 
. ring r5;
> r5;
//   characteristic : 32003
//   number of vars : 3
//        block   1 : ordering dp
//                  : names    x y z 
//        block   2 : ordering C
> setring r;
> poly f = x**3 + y**2;
> poly g = x3+y2;
> f;
x3+y2
> g;
x3+y2
> g + 4xy;
x3+4xy+y2
> f = x3+y3+(x-y)*x2y2+z2; 
> g = f^2*(2x-y);
> g;
2x7y4-5x6y5+4x5y6-x4y7+4x7y2-6x6y3+2x5y4+4x4y5-6x3y6+2x2y7+4x4y2z2-6x3y3z2+2x2y4z2+2x7-x6y+4x4y3-2x3y4+2xy6-y7+4x4z2-2x3yz2+4xy3z2-2y4z2+2xz4-yz4
> ideal I = f,g;
> // Is (1,2,0) on the variety V(I)?
> ideal J = subst(I, var(1), 1);
> J = subst(J, var(2), 2);
> J = subst(J, var(3), 0);
> J;
J[1]=2
J[2]=4
> // Is g contained in ideal generated by f?
> ideal gbI = groebner(f);
> reduce(g,gbI);
0


> // play with Jacobian ideal of f
. ideal J = jacob(f);
// ** redefining J **
> J;
J[1]=3x2y2-2xy3+3x2
J[2]=2x3y-3x2y2+3y2
J[3]=2z
> J = groebner(J);
> J;
J[1]=z
J[2]=3x2y2-2xy3+3x2
J[3]=2x3y-2xy3+3x2+3y2
J[4]=4x4-12xy3+4y4+9x2+9y2
J[5]=20y5+54x3-45x2y-30xy2-81y3
J[6]=10xy4+18x3-15x2y-27y3
> ideal K = kbase(J);
> K;
K[1]=y4
K[2]=xy3
K[3]=y3
K[4]=xy2
K[5]=y2
K[6]=x2y
K[7]=xy
K[8]=y
K[9]=x3
K[10]=x2
K[11]=x
K[12]=1
> size(K);
12
> vdim(J);
12
> // local orderings
. 
. ring r=0, (x,y,z), ds;
// ** redefining r **
> poly s1 = x3-yz;
> poly s2 = y3-xz;
> poly s3 = z3-xy;
> ideal I = s1,s2,s3;
> ideal J=std(I);
> mult(J);
11
> kbase(J);
_[1]=z4
_[2]=z3
_[3]=z2
_[4]=z
_[5]=y3
_[6]=y2
_[7]=y
_[8]=x3
_[9]=x2
_[10]=x
_[11]=1
> 
. // solve library 
. LIB "solve.lib";
// ** loaded /usr/local/Singular/2-0-4/LIB/solve.lib (1.21.2.13,2002/10/21)
// ** loaded /usr/local/Singular/2-0-4/LIB/triang.lib (1.7,2001/02/19)
// ** loaded /usr/local/Singular/2-0-4/LIB/elim.lib (1.14.2.3,2002/10/24)
// ** loaded /usr/local/Singular/2-0-4/LIB/poly.lib (1.33.2.5,2002/04/09)
// ** loaded /usr/local/Singular/2-0-4/LIB/ring.lib (1.17.2.1,2002/02/20)
// ** loaded /usr/local/Singular/2-0-4/LIB/inout.lib (1.21.2.5,2002/06/12)
// ** loaded /usr/local/Singular/2-0-4/LIB/general.lib (1.38.2.8,2002/04/30)
// ** loaded /usr/local/Singular/2-0-4/LIB/matrix.lib (1.26.2.2,2002/10/07)
// ** loaded /usr/local/Singular/2-0-4/LIB/random.lib (1.16.2.1,2002/02/20)
> list s = solve(I);
// name of new current ring: rC
> s[1];
[1]:
   -1
[2]:
   i
[3]:
   i
> // THE EXERCISE (a typo corrected in the second generator) 
> ring r=0, (x,y), ds;
// ** redefining r **
> ideal I = x2-2x+y2, x2-4x+3y4;
> list s = solve(I);
// name of current ring: rC
> s[7]; // take one of the integer solutions
[1]:
   1
[2]:
   1
> // PART (d)
> setring r; // return to the old ring
> ideal J = subst(I, var(1), var(1)+1);
// ** redefining J **
> J = subst(J, var(2), var(2)+1);;
> J;
J[1]=2y+x2+y2
J[2]=-2x+12y+x2+18y2+12y3+3y4
> mult(std(J));
1
>