// Anelli di polinomi: coefficienti, variabili, ordine monomiale
// ad esemipio R = Q[x,y,z]
;
ring R = 0, (x,y,z), lp;
R;
//   characteristic : 0
//   number of vars : 3
//        block   1 : ordering lp
//                  : names    x y z
//        block   2 : ordering C
number a = 2;
a;
2
number b = 2/3;
b;
2/3
poly f1 = x^2+y+z-1;
poly f2 = x+y^2+z-1;
poly f3 = x+y+z^2-1;
ideal I = f1, f2, f3;
groebner(I);
_[1]=z6-4z4+4z3-z2
_[2]=2yz2+z4-z2
_[3]=y2-y-z2+z
_[4]=x+y2+z-1
// si veda [CLO], pag. 115;
;
ideal GI = _;
GI;
GI[1]=z6-4z4+4z3-z2
GI[2]=2yz2+z4-z2
GI[3]=y2-y-z2+z
GI[4]=x+y2+z-1
poly g_1 = GI[4];
g_1;
x+y2+z-1
std(I);
_[1]=z6-4z4+4z3-z2
_[2]=2yz2+z4-z2
_[3]=y2-y-z2+z
_[4]=x+y+z2-1
// g_1 di [CLO] e' il quarto polinomio
;
option(redSB) // option della base do Groebner ridotta
;
groebner(I);
_[1]=z6-4z4+4z3-z2
_[2]=2yz2+z4-z2
_[3]=y2-y-z2+z
_[4]=x+y+z2-1
groebner(GI);
_[1]=z6-4z4+4z3-z2
_[2]=2yz2+z4-z2
_[3]=y2-y-z2+z
_[4]=x+y+z2-1
// ideali di eliminazione;
;
ideal G = groebner(I);
ideal I1 = G[1], G[2], G[3];
I1;
I1[1]=z6-4z4+4z3-z2
I1[2]=2yz2+z4-z2
I1[3]=y2-y-z2+z
ideal I2 = G[1];
LIB "elim.lib";
// ** loaded /usr/share/Singular/LIB/elim.lib (14661,2012-03-05)
// ** loaded /usr/share/Singular/LIB/ring.lib (15322,2012-10-12)
// ** loaded /usr/share/Singular/LIB/primdec.lib (14732,2012-03-30)
// ** loaded /usr/share/Singular/LIB/absfact.lib (14191,2011-05-04)
// ** loaded /usr/share/Singular/LIB/triang.lib (13499,2010-10-15)
// ** loaded /usr/share/Singular/LIB/matrix.lib (13658,2010-11-16)
// ** loaded /usr/share/Singular/LIB/nctools.lib (14246,2011-05-26)
// ** loaded /usr/share/Singular/LIB/random.lib (14661,2012-03-05)
// ** loaded /usr/share/Singular/LIB/poly.lib (14852,2012-04-30)
// ** loaded /usr/share/Singular/LIB/general.lib (14191,2011-05-04)
// ** loaded /usr/share/Singular/LIB/inout.lib (13499,2010-10-15)
ideal i1 = elim(I,(1));
i1;
i1[1]=y2-y-z2+z
i1[2]=2yz2+z4-z2
I1;
I1[1]=z6-4z4+4z3-z2
I1[2]=2yz2+z4-z2
I1[3]=y2-y-z2+z
groebner(i1);
_[1]=z6-4z4+4z3-z2
_[2]=2yz2+z4-z2
_[3]=y2-y-z2+z
LIB "teachstd.lib";
// ** loaded /usr/share/Singular/LIB/teachstd.lib (13499,2010-10-15)
// S-polinomio sulla pagina 84 di [CLO]
;
ring R2 = 0, (x,y), Dp;
poly f=x^3*y^2-x^2*y^3+x; poly g=3*x^4*y+y^2;
spoly(f,g);
-x3y3-1/3y3+x2
//cosa e piu' grande? x*2*y oppur y^4 ?
;
y^4 > x^2*y;
1
y^4 < x^2*y;
0
y^4 <> x^2*y;
1
y^4 != x^2*y;
1
y^4 == x^2*y;
0
ring R3 = (real, 10), (x,y), Dp;
f;
   ? `f` is undefined
   ? error occurred in or before STDIN line 57: `f;`
poly ff = fetch(R2,f);
ff;
x3y2-x2y3+x
poly gg = fetch(R2,g);
spoly(ff,gg);
-x3y3-0.3333333333*y3+x2
ring R4 = (real, 100), (x,y), Dp;
poly f100 = fetch(R2,f);
poly g100 = fetch(R2,g);
spoly(f100,g100);
-x3y3-0.3333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333*y3+x2
ring R5 = (real,a,10), (x,y), Dp;
   ? too many parameters
   ? cannot make ring
   ? error occurred in or before STDIN line 66: `ring R5 = (real,a,10), (x,y), Dp;`
   ? expected ring-expression. type 'help ring;'
ring R5 = ((real,10),a), (x,y), Dp;
poly fp = (a+0.3)*x+a/(a+0.23)*x*y;
fp;
(0.9497578118+a*0.2184442967)*xy+(0.3+a*1)*x
number n = a/(a+0.23);
n;
(0.9497578118+a*0.2184442967)
number n = a/(a+1);
// ** redefining n **
n;
(0.5+a*0.5)
R5;
//   characteristic : 0 (complex:10 digits, additional 10 digits)
//   1 parameter    : a 
//   minpoly        : (a^2+1)
//   number of vars : 2
//        block   1 : ordering Dp
//                  : names    x y
//        block   2 : ordering C
help ring;
// ** Displaying help in browser 'htmlhelp'.
// ** Use 'system("--browser", <browser>);' to change browser,
// ** where <browser> can be: "htmlhelp", "xinfo", "info", "builtin", "dummy", "emacs".
ring R6 = complex, (x), lp;
i^2;
-1
number n = 0.33*i;
number n = i/3;
// ** redefining n **
n;
(i*0.333333)
ring R6 = (real,a), (x), lp;
// ** redefining R6 **
R6;
//   characteristic : 0 (complex:6 digits, additional 6 digits)
//   1 parameter    : a 
//   minpoly        : (a^2+1)
//   number of vars : 1
//        block   1 : ordering lp
//                  : names    x
//        block   2 : ordering C
ring R = (2, a),(x,y), lp;
// ** redefining R **
minpoly=a^2+a+1;
R;
//   characteristic : 2
//   1 parameter    : a 
//   minpoly        : (a2+a+1)
//   number of vars : 2
//        block   1 : ordering lp
//                  : names    x y
//        block   2 : ordering C
monitor("");