LIB "tst.lib"; tst_init(); 2+3+4; //-> 9 2+3; 3+5; //-> 5 //-> 8 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 +1+1+1+1+1+1+1+1+1+1+1+1+1+1+1; //-> 42 1/3 + 1/5; //-> ? no ring active //-> ? error occurred in STDIN line ...: `1/3 + 1/5;` ring R = 0, (x,y), dp; ring R1 = 0, x(1..7), dp; ring R2 = (0,i), (x,y), dp; minpoly = i^2+1; ring R3 = (2,a), (x,y), dp; minpoly = a^3+a+1; ring R3prime = (2^3,a), (x,y), dp; ring R4 = (32003,s,t), (x,y), dp; //====================== Example 3.2 ============================= ring R5 = 0, (w,x,y,z), dp; matrix m[2][3] = w,x,y,x,y,z; ideal I = minor(m,2); qring Q = groebner(I); basering; //-> // characteristic : 0 //-> // number of vars : 4 //-> // block 1 : ordering dp //-> // : names w x y z //-> // block 2 : ordering C //-> // quotient ring from ideal //-> _[1]=y2-xz //-> _[2]=xy-wz //-> _[3]=x2-wy setring R2; 1/3+1/5; //-> 8/15 setring R3; 1/3+1/5; //-> 0 varstr(R3); //-> x,y poly f = x+y; f; //-> x+y ring S3 = (2^3,a), (y,x), dp; poly f = x+y; f; //-> y+x kill R,R1,R2,R3,R3prime,R4,R5,Q,S3; //====================== new Session ============================== ring R = 0, x(1..7), (dp(3),wp(2,1),dp); basering; //-> // characteristic : 0 //-> // number of vars : 7 //-> // block 1 : ordering dp //-> // : names x(1) x(2) x(3) //-> // block 2 : ordering wp //-> // : names x(4) x(5) //-> // : weights 2 1 //-> // block 3 : ordering dp //-> // : names x(6) x(7) //-> // block 4 : ordering C kill R; ring R = 0, (x,y,z), M(1,1,1, 0,0,-1, 0,-1,0); kill R; ring R = 0, (x,y,z,w), (a(1,0,0,1),dp); kill R; //====================== Example 3.8 (new Session) ================ ring R = 0, (x,y), (c,dp); kill R; //====================== Remark 3.9 (new Session) ================= ring R = 0, (x,y,e(1..3)), wp(1,1,-1,-2,-4); ideal I = e(1..3); qring Q = groebner(I^2); kill Q,R; //====================== new Session ============================= ring R = 0, (x,y), dp; int i = 1; // object of ring independent type int poly f = x; // object of ring dependent type poly ring S = 0, (x,y), dp; // active ring is changed poly g = y; f; //-> ? `f` is undefined //-> ? error occurred in STDIN line 6: `f;` listvar(); //-> // S [0] *ring //-> // g [0] poly //-> // i [0] int 1 //-> // R [0] ring kill S,i,R; //====================== Example 3.10 (new Session) ============== ring R = 0, (x,y), dp; poly f = x2+y; ring S = 0, (a,b,c), dp; map F = R, a-b, c; // map F: R->S, sending x to a-b, y to c poly g = F(f); // apply the map g; //-> a2-2ab+b2+c ring S1 = 2, (a,b,c), lp; qring Q = std(a^2); map F1 = R, a-b, c; // target ring is qring, with another // characteristic and monomial order poly g=F1(f); g; // polynomial is not yet reduced //-> a2+b2+c reduce(g,std(0)); //-> b2+c ring R1 = 0, (a,b,c,x,y,z), dp; fetch(R,f); // fetch preserves order of variables //-> a2+b imap(R,f); // imap preserves names of variables //-> x2+y fetch(Q,g); //-> a2+b2+c kill R,S,S1,R1,Q; //====================== new Session ============================= ring R = 0, (x,y,z), dp; ideal I = x2-y, y4-z2; kill R; //====================== new Session ============================= ring R = 0, (x,y,z), dp; poly f = x2-y; poly g = y4-z2; ideal I = f,g; vector v = [f,0,0,g,0]; kill v; // ---- Alternatively: vector v = f*gen(1)+g*gen(4); kill R; ring R = 0, (x,y,z), (c,dp); vector v = [x,y]+[x2,1,z3,0]; v; //-> [x2+x,y+1,z3] ring S = 0, (x,y,z), (dp,c); vector v = fetch(R,v); v; //-> z3*gen(3)+x2*gen(1)+x*gen(1)+y*gen(2)+gen(2) print(v); //-> [x2+x,y+1,z3] kill S,R; //====================== new Session ============================= ring R = 0, (x,y), dp; module I = [x2,-y,y,-y,0], [0,0,y], [y,x]; print(I); //-> x2,0,y, //-> -y,0,x, //-> y, y,0, //-> -y,0,0 matrix MI[4][3] = x2, 0, y, -y, 0, x, y, y, 0, -y, 0, 0; kill MI; // ---- Alternatively: matrix MI = I; kill R; //====================== Section 3.4 (new Session) ================= ring R = 0, (w,x,y,z), dp; module I = [xz,0,-w,-1,0], [-yz2,y2, 0,-w,0], [y2z,0,-z2,0,-x], [y3,0,-yz,-x,0], [-z3,yz,0,0,-w], [-yz2,y2,0,-w,0], [0,0,-wy2+xz2,-y2,x2]; print(I); //-> xz,-yz2,y2z,y3, -z3,-yz2,0, //-> 0, y2, 0, 0, yz, y2, 0, //-> -w,0, -z2,-yz,0, 0, -wy2+xz2, //-> -1,-w, 0, -x, 0, -w, -y2, //-> 0, 0, -x, 0, -w, 0, x2 attrib(I,"isHomog"); // no attribute => empty output homog(I); //-> 1 attrib(I,"isHomog"); //-> 0,1,1,2,2 print(betti(I,0),"betti"); //-> 0 1 //-> ------------------ //-> 0: 1 - //-> 1: 2 1 //-> 2: 2 5 //-> 3: - 1 //-> ------------------ //-> total: 5 7 intvec DV = 2,3,3,4,4; attrib(I,"isHomog",DV); attrib(I,"isHomog"); //-> 2,3,3,4,4 print(betti(I,0),"betti"); //-> 0 1 //-> ------------------ //-> 2: 1 - //-> 3: 2 1 //-> 4: 2 5 //-> 5: - 1 //-> ------------------ //-> total: 5 7 intmat BI = betti(I,0); int d = attrib(BI,"rowShift"); d; //-> 2 kill R,DV,BI,d; //====================== Section 3.5 (new Session) ================= ring R = 0, (x,y,z), lp; ideal I = 2y+z,3x-y; std(I); //-> _[1]=2y+z //-> _[2]=3x-y option(redSB); ideal G = std(I); G; //-> G[1]=2y+z //-> G[2]=6x+z G = simplify(G,1); G; //-> G[1]=y+1/2z //-> G[2]=x+1/6z kill R; //====================== Example 3.13 (new Session) =============== ring R = 0, (x,y,z), dp; ideal I = 3x3y+x3+xy3+y2z2, 2x3z-xy-xz3-y4-z2, 2x2yz-2xy2+xz2-y4; option(redSB); // force computation of reduced GBs int aa = timer; ideal SI = std(I); size(SI); dim(SI); ring S = 0, (x,y,z), lp; aa = timer; ideal J = fglm(R,SI); size(J); // number of generators //-> 8 size(string(J))/68; // number of lines with 68 characters // needed to display J: //-> 631 deg(J[1..size(J)]); // degrees of the generators //-> 35 34 34 34 34 34 34 34 leadmonom(J[1..size(J)]); // generators for L(I) w.r.t. lp //-> z35 yz6 y2z4 y3z2 y5 xz2 xy x3 leadcoef(J[8]); // leading coefficient of 8th generator //-> 6440093631623773985969509841859276602512807348986590906348582267651806942677443883093109641441627364249598438582596862938314965556548533870597328962260825040847335705757819599104 ideal I = fetch(R,I); // I = std(I); //-> error: no more memory kill S,aa,R; //====================== Example 3.14 (new Session) =============== LIB "grwalk.lib"; ring S = 0, (x,y,z), lp; ideal I = 3x3y+x3+xy3+y2z2, 2x3z-xy-xz3-y4-z2, 2x2yz-2xy2+xz2-y4; option(redSB); // force computation of reduced GBs int aa = timer; ideal J = fwalk(I); size(J), dim(J); kill S,aa; //====================== Example 3.16 (new Session) =============== ring S = 0, (x,y,z), lp; ideal I = 3x3y+x3+xy3+y2z2, 2x3z-xy-xz3-y4-z2, 2x2yz-2xy2+xz2-y4; option(redSB); ring Rhom = 0, (x,y,z,t), dp; ideal I = imap(S,I); ideal Ih = homog(I,t); // generators of I are homogenized int aa = timer; Ih = std(Ih); intvec H = hilb(Ih,1); ring Shom = 0, (x,y,z,t), lp; ideal Ih = imap(Rhom,Ih); Ih = std(Ih,H); Ih = subst(Ih,t,1); setring S; ideal J = imap(Shom,Ih); size(J); //-> 102 J = interred(J); //-> 3 size(J); //-> 8 dim(J); kill S,Rhom,aa,H,Shom; //====================== Remark 3.18 (new Session) =============== ring R = (32003,a,b,c,d), (t,u,v,w,x,y,z), dp; ideal I = -cw+bx, ct+2au-2bu-2cv-(ad+bd), -2tx+4wy+4xz+ct-2aw-2dw-2by-2cz+(ab+bd), t*(z-x)+(a-b+d)*(y-w)+c*(x-z), -tw+a*(t-x)+dx, -2tv+ct-2du+(ad+bd), ct2-(b2-ab+c2)*t-(acd-cd2); int aa = timer; ideal SI = slimgb(I); size(SI), dim(SI); SI = std(I); size(SI), dim(SI); kill R,aa; //====================== Example 3.20 (new Session) =============== ring S = 0, (x,y,z), lp; ideal I = 3x3y+x3+xy3+y2z2, 2x3z-xy-xz3-y4-z2, 2x2yz-2xy2+xz2-y4; option(redSB); option(prot); int aa = timer; ideal J = groebner(I); //-> std in (0),(x,y,z,@t),(dp,C) //-> [255:1]4(2)sss5s6s7(3)s(4)s(5)s(6)s8(8)s(9)-ss(11)s(12)---9-s(9)-s( //-> 10)--s--10-s(8)s(9)-s---11------ //-> product criterion:9 chain criterion:124 //-> std with hilb in (0),(x,y,z,@t),(lp(3),C) //-> [255:1]4(2)sss5ss6s(3)s(5)s7(6)s(7)s(9)s(11)s(13)-s(14)s8(16)s(17)s //-> (19)s(21)s(23)s(25)s(27)s(28)-s(29)--shhhhh9(24)s(26)s(28)s(30)s(32 //-> )s(33)s(35)s(37)s(39)s(41)shhhhhhhhhhhhhhh10(28)ss(29)s(30)s(32)s(3 //-> 4)s(35)s(37)s(39)s(41)s(43)shhhhhhhhhhhhhhhhhh11(26)s(28)s(30)s(32) //-> s(34)s(35)shhhhhhhhhhhhhhhhhhhhh12(16)s(18)s(20)s(22)s(24)shhhhhhhh //-> hhhh13(14)s(15)s(17)s(19)s(21)shhhhhhhhhh14(13)s(15)s(17)s(19)shhhh //-> hhhhhh15(10)s(12)s(14)shhhhhhhh16(8)s(10)s(12)shhhhhh17(8)s(10)s(12 //-> )shhhhhh18(8)s(9)s(11)shhhhhh19(7)s(9)shhhhhh20(5)s(7)shhhh21(5)s(7 //-> )shhhh22(5)s(7)shhhh23(5)s(7)shhhh24(5)s(6)shhhh25(4)shhhh26(2)shh2 //-> 7shh28shh29shh30shh31shh32shh33shh34shh35shh36shh37shh38shhhh //-> product criterion:27 chain criterion:4846 //-> hilbert series criterion:175 //-> dehomogenization //-> imap to original ring //-> simplification //-> interreduction size(J); //-> 8 kill S,aa; option(noprot); //====================== Example 3.21 (new Session) =============== ring R = 0, (x,y), dp; ideal I = 4x2y2+3x, y3+2xy, 7x3+6y; std(I); //-> _[1]=y //-> _[2]=x ideal J = x; matrix A = lift(I,J); A; //-> A[1,1]=-3670016/18809541x2y+9604/6269847xy2-134217728/131666787y3-128/63xy-100352/6269847y2-458752/6269847y+1/3 //-> A[2,1]=536870912/131666787x2y2+401408/6269847x2y+1835008/6269847x2-4194304/6269847xy+10976/2089949y2+64/21x+50176/2089949y+229376/2089949 //-> A[3,1]=2097152/18809541xy3-5488/6269847y4-25088/6269847y3-114688/6269847y2 matrix(I)*A; //-> _[1,1]=x kill R; //====================== Example 3.22 (new Session) =============== ring R = 0, (x,y), dp; ideal I = x7+x5y2, y4-xy7; poly f1, f2 = x6y7+x3y5, x6y7+x7y2; ideal GI = groebner(I); reduce(f1,GI,1); // see Example 1.39 for reduce //-> y5-y4 reduce(f2,GI,1); //-> 0 lift(I,f1); //-> ? 2nd module lies not in the first //-> ? error occurred in STDIN line 8: `lift(I,f1); ` matrix C = lift(I,f2); C; //-> C[1,1]=x4y22-x2y24-x3y19+xy21+y2 //-> C[2,1]=x10y15-x6y19-x5 f2 - C[1,1]*I[1] - C[2,1]*I[2]; // check (result must be 0) //-> 0 ideal J1 = f1, f2; ideal J2 = f2, x5y9+x6y4; reduce(J1,GI,1); // normal form for the generators of J1 //-> _[1]=y5-y4 //-> _[2]=0 size(reduce(J2,GI,1)); //-> 0 kill R; //====================== Example 3.23 (new Session) =============== ring R = 0, (x,y), dp; ideal I = maxideal(3); // the ideal ^3 poly f1, f2 = x, 1-x; ring S = 0, (x,y,t), dp; ideal I = imap(R,I); poly f1 = imap(R,f1); ideal Jf1 = I, t*f1-1; Jf1 = std(Jf1); reduce(1,Jf1,1); // result is 0 iff f1 is in radical(I) //-> 0 poly f2 = imap(R,f2); ideal Jf2 = I, t*f2-1; Jf2 = std(Jf2); reduce(1,Jf2,1); //-> 1 setring R; // ---- Alternatively --------- if (not(defined(rad_con))){ LIB "poly.lib"; } rad_con(f1,I); // result is 1 iff f is in radical(I) //-> 1 rad_con(f2,I); //-> 0 kill R,S; //====================== Example 3.24 (new Session) =============== ring R = 0, (a,b,c,d,e,f,g,t,u,v,w,y,z), dp; ideal I = z2+e2-1, g2+w2+a2-1, t2+u2+b2-1, f2+v2+c2-1, y2+d2-1, zw+ea, gt+wu+ab, tf+uv+bc, fy+cd, a+b+c+d+e, f+g+t+y+1, u+v+w+z-1; ring Rhom = 0, (a,b,c,d,e,f,g,t,u,v,w,y,z,h), dp; ideal I = imap(R,I); ideal J = homog(I,h); // homogenize the given generators int aa = timer; ideal L = std(J); intvec H = hilb(L,1); // assign Hilbert series ideal K = eliminate(J,abcdefgtuvw,H); K = subst(K,h,1); // dehomogenize size(K); //-> 1 K[1]; // the equation //-> 790272y16z16-3612672y16z15+3612672y15z16-6530048y16z14-6006784y15z15 //-> -6530048y14z16+41607168y16z13-56159232y15z14+[...] kill R,Rhom,aa,H; //====================== Example 3.27 (new Session) =============== ring R = 0, x(1..3), dp; poly f1 = x(1)^6*x(3)^2-x(2)^6*x(3)^2; poly f2,f3,f4 = x(1)^3-x(2)^3, x(1)^3+x(2)^3, x(3)^3; ring S = 0, y(1..4), dp; setring R; ideal zero; // the zero ideal map phi = S,f1,f2,f3,f4; setring S; preimage(R,phi,zero); // the kernel of phi //-> _[1]=y(2)^3*y(3)^3*y(4)^2-y(1)^3 setring R; //--- Alternatively: kill S; if (not(defined(algDependent))){ LIB "algebra.lib"; } list L = algDependent(ideal(f1,f2,f3,f4)); L[1]; // first entry of L is 1 iff the polynomials are // algebraically dependent //-> 1 def S = L[2]; // second entry of L is a ring which contains // an ideal ker defining the algebraic relation setring S; ker; //-> ker[1]=y(2)^3*y(3)^3*y(4)^2-y(1)^3 kill R,S,L; //====================== Example 3.28 (new Session) =============== ring R = 0, x(1..3), dp; poly f = x(1)^6*x(2)^6-x(1)^6*x(3)^6; poly f1 = x(1)^3*x(2)^3-x(1)^3*x(3)^3; poly f2 = x(1)^3*x(2)^3+x(1)^3*x(3)^3; ring S = 0, (x(1..3),y(1..2)), (dp(3),dp(2)); ideal J = imap(R,f1)-y(1), imap(R,f2)-y(2); ideal G = groebner(J); reduce(imap(R,f),G); //-> y(1)*y(2) setring R; kill S; //--- Alternatively: if (not(defined(algebra_containment))){ LIB "algebra.lib"; } algebra_containment(f,ideal(f1,f2)); //-> // y(1)*y(2) //-> 1 def L = algebra_containment(f,ideal(f1,f2),1); def S = L[2]; setring S; check; // polynomial defining the algebraic relation //-> y(1)*y(2) kill R,S,L; //====================== Example 3.29 (new Session) =============== ring C = 0, (a,b,c,x,y,z), (dp(3),dp); ideal J = c-b3, 2a+b6-x, 7b-a2-y, c2-z; option(redSB); simplify(groebner(J),1); // the reduced Groebner basis for J //-> _[1]=x12-12x11z+66x10z2-220x9z3+495x8z4-792x7z5+[...] //-> _[2]=c-1/21952x6+3/10976x5z-15/21952x4z2+[...] //-> _[3]=b-1/28x2+1/14xz-1/28z2-1/7y //-> _[4]=a-1/2x+1/2z if (not(defined(is_surjective))){ LIB "algebra.lib"; } ring B = 0, (x,y,z), dp; ring A = 0, (a,b,c), dp; qring Q = groebner(c-b3); // quotient ring map psi = B, 2a+b6, 7b-a2, c2; is_surjective(psi); //-> 1 kill A,B,C,Q; //====================== Example 3.30 (new Session) =============== ring R = 0, (w,x,y,z), dp; poly f1, f2, f3 = y2-xz, xy-wz, x2z-wyz; ideal I = f1, f2, f3; module phi2 = syz(I); print(phi2); //-> x, wz, //-> -y,-xz, //-> 1, y size(syz(phi2)); // we check that there are no higher syzygies //-> 0 resolution FI = nres(I,0); typeof(FI[1]); // 'typeof' displays type of given object //-> ideal print(FI[1]); //-> y2-xz, //-> xy-wz //-> x2z-wyz typeof(FI[2]); //-> module print(FI[2]); //-> x, wz, //-> -y,-xz, //-> 1, y print(betti(FI),"betti"); //-> 0 1 2 //-> ------------------------ //-> 0: 1 - - //-> 1: - 2 - //-> 2: - - 1 //-> ------------------------ //-> total: 1 2 1 print(betti(FI,0),"betti"); //-> 0 1 2 //-> ------------------------ //-> 0: 1 - - //-> 1: - 2 1 //-> 2: - 1 1 //-> ------------------------ //-> total: 1 3 2 kill R; //====================== Example 3.35 (new Session) =============== ring R = 0, (w,x,y,z), dp; module I = [xz,0,-w,-1,0], [-yz2,y2, 0,-w,0], [y2z,0,-z2,0,-x], [y3,0,-yz,-x,0], [-z3,yz,0,0,-w], [-yz2,y2,0,-w,0], [0,0,-wy2+xz2,-y2,x2]; print(I); //-> xz,-yz2,y2z,y3, -z3,-yz2,0, //-> 0, y2, 0, 0, yz, y2, 0, //-> -w,0, -z2,-yz,0, 0, -wy2+xz2, //-> -1,-w, 0, -x, 0, -w, -y2, //-> 0, 0, -x, 0, -w, 0, x2 homog(I); //-> 1 attrib(I,"isHomog"); //-> 0,1,1,2,2 resolution FInres = nres(I,0); print(betti(FInres,0),"betti"); //-> 0 1 2 //-> ------------------------ //-> 0: 1 - - //-> 1: 2 1 1 //-> 2: 2 5 1 //-> 3: - 1 1 //-> ------------------------ //-> total: 5 7 3 print(FInres[1]); // the given generators //-> xz,-yz2,y2z,y3, -z3,-yz2,0, //-> 0, y2, 0, 0, yz, y2, 0, //-> -w,0, -z2,-yz,0, 0, -wy2+xz2, //-> -1,-w, 0, -x, 0, -w, -y2, //-> 0, 0, -x, 0, -w, 0, x2 print(FInres[2]); // display syzygies on the given generators //-> 0, y2,0, //-> -1,0, xz, //-> 0, -x,wy, //-> 0, 0, -wz, //-> 0, 0, -xy, //-> 1, 0, 0, //-> 0, -1,0 size((FInres[3])); //-> 0 resolution FImres = mres(I,0); print(betti(FImres,0),"betti"); //-> 0 1 2 //-> ------------------------ //-> 0: 1 - - //-> 1: 2 1 - //-> 2: 2 4 - //-> 3: - - 1 //-> ------------------------ //-> total: 5 5 1 print(FImres[1]); // the new generators //-> xz,z3, yz2,y2z,y3, //-> 0, -yz,-y2,0, 0, //-> -w,0, 0, -z2,-yz, //-> -1,0, w, 0, -x, //-> 0, w, 0, -x, 0 print(FImres[2]); // display syzygies on the new generators //-> 0, //-> xy, //-> -xz, //-> wy, //-> -wz module PI = prune(I); print(betti(PI,0),"betti"); //-> 0 1 //-> ------------------ //-> 0: 1 - //-> 1: 2 - //-> 2: 1 5 //-> 3: - 1 //-> ------------------ //-> total: 4 6 print(PI); //-> wxz+yz2,-y2z,-y3+x2z,z3, wxz+yz2,xy2z, //-> -y2, 0, 0, -yz,-y2, 0, //-> -w2, z2, -wx+yz, 0, -w2, -xz2, //-> 0, x, 0, w, 0, -x2 resolution FPImres = mres(PI,0); print(betti(FPImres,0),"betti"); //-> 0 1 2 //-> ------------------------ //-> 0: 1 - - //-> 1: 2 - - //-> 2: 1 4 - //-> 3: - - 1 //-> ------------------------ //-> total: 4 4 1 print(FPImres[1]); //-> z3, y2z,wxz+yz2,y3-x2z, //-> -yz,0, -y2, 0, //-> 0, -z2,-w2, wx-yz, //-> w, -x, 0, 0 print(FPImres[2]); //-> xy, //-> wy, //-> -xz, //-> -wz resolution FPIsres = sres(PI,0); //-> ? ideal not a standardbasis //-> ? error occurred in STDIN line 27: //-> `resolution FPIsres = sres(PI,0);` resolution FPIsres = sres(groebner(PI),0); print(betti(FPIsres,0),"betti"); //-> 0 1 2 3 4 //-> ------------------------------------ //-> 0: 1 - - - - //-> 1: 2 - - - - //-> 2: 1 4 1 - - //-> 3: - 1 5 1 - //-> 4: - 4 2 3 1 //-> 5: - 1 3 1 - //-> ------------------------------------ //-> total: 4 10 11 5 1 resolution FInresmin = minres(FInres); print(betti(FInresmin,0),"betti"); //-> 0 1 2 //-> ------------------------ //-> 0: 1 - - //-> 1: 2 1 - //-> 2: 2 4 - //-> 3: - - 1 //-> ------------------------ //-> total: 5 5 1 resolution FPIsresmin = minres(FPIsres); print(betti(FPIsresmin,0),"betti"); //-> 0 1 2 //-> ------------------------ //-> 0: 1 - - //-> 1: 2 - - //-> 2: 1 4 - //-> 3: - - 1 //-> ------------------------ //-> total: 4 4 1 kill R; listvar(); //====================== Example 3.40(1) (new Session) =============== ring D3 = 0, (x(1..3),d(1..3)), dp; int i,j; matrix C[6][6]; for (i=1; i<=6; i++) { for (j=i+1; j<=6; j++) { C[i,j] = 1; } } matrix H[6][6]; H[1,4] = 1; H[2,5] = 1; H[3,6] = 1; def @S=nc_algebra(C,H); setring @S; kill D3; // ---- Alternatively: if (not(defined(Exterior))){ LIB "nctools.lib"; } ring D3 = 0, (x(1..3),d(1..3)), dp; def @S3=Weyl(); setring @S3; basering; //-> // characteristic : 0 //-> // number of vars : 6 //-> // block 1 : ordering dp //-> // : names x(1) x(2) x(3) d(1) d(2) d(3) //-> // block 2 : ordering C //-> // noncommutative relations: //-> // d(1)x(1)=x(1)*d(1)+1 //-> // d(2)x(2)=x(2)*d(2)+1 //-> // d(3)x(3)=x(3)*d(3)+1 //================= Example 3.44 (continued Session) =============== ideal I = x(1)^2*d(2)^2+x(2)^2*d(3)^2, x(1)*d(2)+x(3); option(redSB); ideal LSI = std(I); LSI; //-> LSI[1]=x(1)*d(2)+x(3) //-> LSI[2]=x(3)^2 //-> LSI[3]=x(2)*x(3)-x(1) //-> LSI[4]=x(1)*x(3) //-> LSI[5]=x(2)^2 //-> LSI[6]=x(1)*x(2) //-> LSI[7]=x(1)^2 def D3_opp = opposite(@S3); setring D3_opp; // active ring is the opposite algebra of @S3 basering; //-> // characteristic : 0 //-> // number of vars : 6 //-> // block 1 : ordering a //-> // : names D(3) D(2) D(1) X(3) X(2) X(1) //-> // : weights 1 1 1 1 1 1 //-> // block 2 : ordering ls //-> // : names D(3) D(2) D(1) X(3) X(2) X(1) //-> // block 3 : ordering C //-> // noncommutative relations: //-> // X(3)D(3)=D(3)*X(3)+1 //-> // X(2)D(2)=D(2)*X(2)+1 //-> // X(1)D(1)=D(1)*X(1)+1 ideal I = oppose(@S3,I); // map I to opposite algebra ideal RSI_opp = std(I); setring @S3; ideal RSI = oppose(D3_opp,RSI_opp); RSI; //-> RSI[1]=x(1)*d(2)+x(3) //-> RSI[2]=x(3)^2 //-> RSI[3]=x(2)*x(3)+x(1) //-> RSI[4]=x(1)*x(3) //-> RSI[5]=x(2)^2 //-> RSI[6]=x(1)*x(2) //-> RSI[7]=x(1)^2 size(reduce(RSI,LSI)); //-> 1 ideal SI = twostd(I); SI; //-> SI[1]=1 kill i,j,D3,D3_opp,@S3; //====================== Example 3.40(2) (new Session) =============== ring R = 0, x(1..3), dp; int i,j; matrix C[3][3]; for (i=1; i<=3; i++) { for (j=i+1; j<=3; j++) { C[i,j] = -1; } } matrix H[3][3]; def @SS=nc_algebra(C,H); setring @SS; ideal Q = x(1)^2, x(2)^2, x(3)^2; Q = twostd(Q); // compute two-sided Groebner basis qring E3 = Q; kill E3,R; // ---- Alternatively: if (not(defined(Exterior))){ LIB "nctools.lib"; } ring R = 0, x(1..3), dp; def E3 = Exterior(); setring E3; basering; //-> // characteristic : 0 //-> // number of vars : 3 //-> // block 1 : ordering dp //-> // : names x(1) x(2) x(3) //-> // block 2 : ordering C //-> // noncommutative relations: //-> // x(2)x(1)=-x(1)*x(2) //-> // x(3)x(1)=-x(1)*x(3) //-> // x(3)x(2)=-x(2)*x(3) //-> // quotient ring from ideal //-> _[1]=x(3)^2 //-> _[2]=x(2)^2 //-> _[3]=x(1)^2 //================= Example 3.46 (continued Session) =============== ideal I = maxideal(1); def rI = mres(I,0); //-> // ** full resolution in a qring may be infinite, //-> // setting max length to 5 print(betti(rI),"betti"); //-> 0 1 2 3 4 5 //-> ------------------------------------------ //-> 0: 1 3 6 10 15 21 //-> ------------------------------------------ //-> total: 1 3 6 10 15 21 print(rI[1],""); //-> x(3),x(2),x(1) print(rI[2]); //-> x(3),x(2),0, x(1),0, 0, //-> 0, x(3),x(2),0, x(1),0, //-> 0, 0, 0, x(3),x(2),x(1) kill i,j,E3,R; //====================== Remark 3.52 (new Session) =============== ring R = (0,a), (x(1..3),y(1..2),z(1..2)), (dp(3),wp(2,5),lp); minpoly = a^2+1; qring Q = std(y(1)^2-x(1)); list L = ringlist(Q); size(L); //-> 4 L[1]; //-> [1]: //-> 0 //-> [2]: //-> [1]: //-> a //-> [3]: //-> [1]: //-> [1]: //-> lp //-> [2]: //-> 1 //-> [4]: //-> _[1]=(a^2+1) L[2][7]; //-> z(2) L[3]; //-> [1]: //-> [1]: //-> dp //-> [2]: //-> 1,1,1 //-> [2]: //-> [1]: //-> wp //-> [2]: //-> 2,5 //-> [3]: //-> [1]: //-> lp //-> [2]: //-> 1,1 //-> [4]: //-> [1]: //-> C //-> [2]: //-> 0 L[4]; //-> _[1]=x(1)-y(1)^2 L[1][2][1] = "b"; // new name for the parameter L[2][8] = "w"; // append a new variable with name w L[3][3][2] = intvec(1,1,1); // raise the size of the third block // of the monomial order def S = ring(L); setring S; basering; //-> // characteristic : 0 //-> // 1 parameter : b //-> // minpoly : (b^2+1) //-> // number of vars : 8 //-> // block 1 : ordering dp //-> // : names x(1) x(2) x(3) //-> // block 2 : ordering wp //-> // : names y(1) y(2) //-> // : weights 2 5 //-> // block 3 : ordering lp //-> // : names z(1) z(2) w //-> // block 4 : ordering C //-> // quotient ring from ideal //-> _[1]=x(1)-y(1)^2 tst_status(1);\$