101/3; // commento ; ring R = 0, (x,y), lp; R; // characteristic : 0 // number of vars : 2 // block 1 : ordering lp // : names x y // block 2 : ordering C 101/3; 101/3 ring r = (real,50),x,dp; 101/3; 33.66666666666666666666666666666666666666666666666667 ad esempio: ring r = 0, (x,y,z), M(1, 0, 0, 0, 1, 0, 0, 0, 1); // Q[x,y,z] con ordine lessicografico x > y > z > 1 ; ring r = 0, (x,y,z), M(1, 0, 0, 0, 1, 0, 0, 0, 1); // ** redefining r ** // Q[x,y,z] con ordine lessicografico x > y > z > 1 ; r; // characteristic : 0 // number of vars : 3 // block 1 : ordering M // : names x y z // : weights 1 0 0 // : weights 0 1 0 // : weights 0 0 1 // block 2 : ordering C // algoritmo di divisione ; ring r=0,(x,y),lp; // ** redefining r ** poly f=x*y^2+1; poly f1=x*y+1; poly f2=y+1; ideal F=f; ideal G=f1,f2; list q = division (F,G); q; [1]: _[1,1]=-1 _[2,1]=xy [2]: _[1]=2 [3]: _[1,1]=1 f - q[1][1,1]*f1-q[1][2,1]*f2 - q[2]; \\ q[2] = resto _[1,1]=0 skipping text from `\`; f - q[1][1,1]*f1-q[1][2,1]*f2 - q[2]; _[1,1]=0 poly g = 4*x*y^2+1; lead(g); 4xy2 leadmonom(g); xy2 x^2*y > x*y^2; 1 x^2*y > x^4*y^2; 0 ring r=0,(x,y),Dp; // ** redefining r ** x^2*y > x*y^2; 1 // Esempio nel quale il comando std() non conviene: ; ring R = 0, (x,y,z), lp; // ** redefining R ** ideal I = x^3+x^2*z+x*z, x^3+x*y^2+1, x^2+x*y*z+y^3; groebner(I); _[1]=12z10-8z9+66z8-36z7+113z6-24z5+66z4-2z3+13z2+1 _[2]=35098y-258516z9+38400z8-1272818z7+22716z6-1731817z5-777748z4-726255z3-556285z2-76362z-55112 _[3]=8x+6y6+2y5z-8y5-18y4z-2y4+16y3z2+8y3z+9y3+4y2z+y2-18yz2+19yz-5y-6z-7 option(redSB); // Sorpresa: ring R = 0, (x,y,z), lp; // ** redefining R ** ideal I = x^10+y^8*z^2+z^10, x*y+y^2+z^2; std(I); _[1]=y20+11y18z2+45y16z4+120y14z6+210y12z8+253y10z10+210y8z12+120y6z14+45y4z16+10y2z18+z20 _[2]=xz18-y19-11y17z2-45y15z4-120y13z6-210y11z8-253y9z10-210y7z12-120y5z14-45y3z16-9yz18 _[3]=xy+y2+z2 _[4]=x2z16+y18+11y16z2+45y14z4+120y12z6+210y10z8+253y8z10+210y6z12+120y4z14+44y2z16+8z18 _[5]=x3z14+7xz16-y17-11y15z2-45y13z4-120y11z6-210y9z8-253y7z10-210y5z12-119y3z14-35yz16 _[6]=x4z12+6x2z14+y16+11y14z2+45y12z4+120y10z6+210y8z8+253y6z10+209y4z12+110y2z14+27z16 _[7]=x5z10+5x3z12+20xz14-y15-11y13z2-45y11z4-120y9z6-210y7z8-252y5z10-200y3z12-75yz14 _[8]=x6z8+4x4z10+14x2z12+y14+11y12z2+45y10z4+120y8z6+209y6z8+243y4z10+165y2z12+48z14 _[9]=x7z6+3x5z8+9x3z10+28xz12-y13-11y11z2-45y9z4-119y7z6-200y5z8-208y3z10-90yz12 _[10]=x8z4+2x6z6+5x4z8+14x2z10+y12+11y10z2+44y8z4+110y6z6+165y4z8+133y2z10+42z12 _[11]=x9z2+x7z4+2x5z6+5x3z8+14xz10-y11-10y9z2-35y7z4-75y5z6-90y3z8-43yz10 _[12]=x10+y8z2+z10 ring R = 0, (x,y,z), lp; // ** redefining R ** ideal I = x^3+x^2*z+x*z, x^3+x*y^2+1, x^2+x*y*z+y^3; ideal GI = groebner(I); GI; GI[1]=12z10-8z9+66z8-36z7+113z6-24z5+66z4-2z3+13z2+1 GI[2]=35098y-258516z9+38400z8-1272818z7+22716z6-1731817z5-777748z4-726255z3-556285z2-76362z-55112 GI[3]=35098x+5904z9-9792z8+45364z7-33520z6+109674z5+724z4+84727z3+26468z2+68785z-11217 factorize(GI[1]); [1]: _[1]=1 _[2]=12z10-8z9+66z8-36z7+113z6-24z5+66z4-2z3+13z2+1 [2]: 1,1 ring R = (real,10), (x,y,z), lp; // ** redefining R ** ideal I = x^3+x^2*z+x*z, x^3+x*y^2+1, x^2+x*y*z+y^3; ideal GI = groebner(I); factorize(GI[1]); // S-polinomio: (pag. 85) LIB "teachstd.lib"; // ** loaded /usr/bin/../share/singular/LIB/teachstd.lib (4.0.0.0,Jun_2013) // ** loaded /usr/bin/../share/singular/LIB/poly.lib (4.0.0.0,Jun_2013) // ** loaded /usr/bin/../share/singular/LIB/ring.lib (4.0.2.2,Jan_2016) // ** loaded /usr/bin/../share/singular/LIB/primdec.lib (4.0.2.0,Apr_2015) // ** loaded /usr/bin/../share/singular/LIB/absfact.lib (4.0.0.0,Jun_2013) // ** loaded /usr/bin/../share/singular/LIB/triang.lib (4.0.0.0,Jun_2013) // ** loaded /usr/bin/../share/singular/LIB/matrix.lib (4.0.0.0,Jun_2013) // ** loaded /usr/bin/../share/singular/LIB/nctools.lib (4.0.0.0,Jun_2013) // ** loaded /usr/bin/../share/singular/LIB/random.lib (4.0.0.0,Jun_2013) // ** loaded /usr/bin/../share/singular/LIB/elim.lib (4.0.0.1,Jan_2014) // ** loaded /usr/bin/../share/singular/LIB/inout.lib (4.0.0.0,Jun_2013) // ** loaded /usr/bin/../share/singular/LIB/general.lib (4.0.0.1,Jan_2014) ring r=0,(x,y),lp; // ** redefining r ** poly f=x^3*y^2-x^2*y^3+x; poly g=3*x^4*y+y^2; spoly(f,g); -x3y3+x2-1/3y3 LIB "surf.lib"; // ** loaded /usr/bin/../share/singular/LIB/surf.lib (4.0.0.0,Jun_2013) ring R = 0,(x,y),dp; // ** redefining R ** ideal J = x^3-y^2; plot(J); Close window to exit from `singularsurf`. 0 ring R = 0,(x,y,z),dp; // ** redefining R ** // superficie romana di Steiner 1844 ideal J = x^2*y^2+x^2*z^2+y^2*z^2-17*x*y*z; surfer(J); Close window to exit from `surfer`. 0 ideal J = x^2*y^2+x^2*z^2+y^2*z^2-17*x*y*z; // ** redefining J ** surfer(J); Close window to exit from `surfer`. 0 surf(J); plot(J); Close window to exit from `singularsurf`. 0 monitor("");