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REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... % Author H.-G. Graebe | Univ. Leipzig | Version 28.6.1995 % graebe@informatik.uni-leipzig.de COMMENT This is an example session demonstrating and testing the facilities offered by the commutative algebra package CALI. END COMMENT; algebraic; on echo; off nat; % To make it easier to compare differing output. showtime; Time: 30 ms comment #################################### ### ### ### Introductory Examples ### ### ### #################################### end comment; % Example 1 : Generating ideals of affine and projective points. vars:={t,x,y,z}; vars := {t,x,y,z}$ setring(vars,degreeorder vars,revlex); {{t,x,y,z},{{1,1,1,1}},revlex,{1,1,1,1}}$ mm:=mat((1,1,1,1),(3,2,3,1),(2,1,3,2)); mm := mat((1,1,1,1),(3,2,3,1),(2,1,3,2))$ % The ideal with zero set at the point in A^4 with coordinates % equal to the row vectors of mm : setideal(m1,affine_points mm); {z**2 - 3*z + 2, y**2 - 4*y + 3, t - y + z - 1, 2*x - y + 2*z - 3, y*z - y - 3*z + 3}$ % All parameters are as they should be : dim m1; 0$ degree m1; 3$ groebfactor m1; {{z - 2,y - 3,t - 2,x - 1}, {z - 1,t - 3,y - 3,x - 2}, {z - 1,t - 1,y - 1,x - 1}}$ resolve m1$ bettinumbers m1; {1,5,9,7,2}$ % The ideal with zero set at the point in P^3 with homogeneous % coordinates equal to the row vectors of mm : setideal(m2,proj_points mm); {2*y**2 - 2*x*z - 7*y*z + 7*z**2, 3*t - 2*x - 2*y + z, 2*x**2 - 4*x*z - y*z + 3*z**2, 2*x*y - 4*x*z - 3*y*z + 5*z**2}$ % All parameters as they should be ? dim m2; 1$ degree m2; 3$ groebfactor m2; {{2*y**2 - 2*x*z - 7*y*z + 7*z**2, 3*t - 2*x - 2*y + z, 2*x**2 - 4*x*z - y*z + 3*z**2, 2*x*y - 4*x*z - 3*y*z + 5*z**2}}$ % It seems to be prime ? isprime m2; no$ % Not, of course, but it is known to be unmixed. % Hence we can use easyprimarydecomposition m2; {{{x - 2*z,y - 3*z,t - 3*z}, {y - 3*z,x - 2*z,t - 3*z}}, {{x - z,y - z,t - z}, {y - z,x - z,t - z}}, {{2*x - z,2*y - 3*z,t - z}, {2*y - 3*z,2*x - z,t - z}}}$ % Example 2 : % The affine monomial curve with generic point (t^7,t^9,t^10). setideal(m,affine_monomial_curve({7,9,10},{x,y,z})); {x**3*y - z**3, x**4 - y**2*z, y**3 - x*z**2}$ % The base ring was changed as side effect : getring(); {{x,y,z},{{7,9,10}},revlex,{7,9,10}}$ vars:=first getring m; vars := {x,y,z}$ % Some advanced commutative algebra : % The analytic spread of m. analytic_spread m; 3$ % The Rees ring Rees_R(vars) over R=S/m. rees:=blowup(m,vars,{u,v,w}); rees := {u**2*v*x - w**3, u*v*x**2 - w**2*z, v*x**3 - w*z**2, u**3*x - v**2*w, u**2*x**2 - v*w*y, u*x**3 - w*y**2, - u*w**2 + v**3, v**2*y - w**2*x, v*y**2 - w*x*z, v*z - w*y, u*z - w*x, u*y - v*x, x**3*y - z**3, x**4 - y**2*z, - x*z**2 + y**3}$ % It is multihomogeneous wrt. the degree vectors, constructed during % blow up. Lets compute weighted Hilbert series : setideal(rees,rees)$ weights:=second getring(); weights := {{0,0,0,7,9,10},{7,9,10,0,0,0}}$ weightedhilbertseries(gbasis rees,weights); ( - x**29*y + x**29 - x**20*y + x**20 - x**19*y**11 + x**19*y**10 - x**19*y + x **19 - x**18*y + x**18 - x**10*y**11 + x**10*y**10 - x**10*y + x**10 - x**9*y **21 + x**9*y**20 - x**9*y**11 + x**9*y**9 - x**9*y + x**9 + y**23 - y**22 + y **16 - y**15 + y**14 - y**11 + y**9 - y**8 + y**7 - y + 1)/(x**7*y - x**7 - y + 1)$ % gr_R(vars), the associated graded ring of the irrelevant ideal % over R. The short way. interreduce sub(x=0,y=0,z=0,rees); {w**3,v**2*w, - u*w**2 + v**3}$ % The long (and more general) way. Gives the result in another % embedding. % Restore the base ring, since it was changed by blowup as a side % effect. setring getring m$ assgrad(m,vars,{u,v,w}); {x, y, z, w**3, v**2*w, - u*w**2 + v**3}$ % Comparing the Rees algebra and the symmetric algebra of M : setring getring m$ setideal(rees,blowup({},m,{a,b,c})); { - y**2*a + z**2*b + x**3*c, x*a - y*b - z*c, - y**2*a**2 + z**2*a*b + x**2*y*b*c + x**2*z*c**2, - y**2*a**3 + z**2*a**2*b + x*y**2*b**2*c + 2*x*y*z*b*c**2 + x*z**2*c**3, - y**2*a**4 + z**2*a**3*b + y**3*b**3*c + 3*y**2*z*b**2*c**2 + 3*y*z**2*b*c** 3 + z**3*c**4}$ % Lets test weighted Hilbert series once more : weights:=second getring(); weights := {{0,0,0,30,28,27},{7,9,10,0,0,0}}$ weightedhilbertseries(gbasis rees,weights); (x**58*y**27 + x**30*y**25 - x**30*y**18 - x**30*y**7 - x**28*y**27 + 1)/(x**85 *y**26 - x**85*y**19 - x**85*y**17 - x**85*y**16 + x**85*y**10 + x**85*y**9 + x**85*y**7 - x**85 - x**58*y**26 + x**58*y**19 + x**58*y**17 + x**58*y**16 - x**58*y**10 - x**58*y**9 - x**58*y**7 + x**58 - x**57*y**26 + x**57*y**19 + x**57*y**17 + x**57*y**16 - x**57*y**10 - x**57*y**9 - x**57*y**7 + x**57 - x**55*y**26 + x**55*y**19 + x**55*y**17 + x**55*y**16 - x**55*y**10 - x** 55*y**9 - x**55*y**7 + x**55 + x**30*y**26 - x**30*y**19 - x**30*y**17 - x** 30*y**16 + x**30*y**10 + x**30*y**9 + x**30*y**7 - x**30 + x**28*y**26 - x** 28*y**19 - x**28*y**17 - x**28*y**16 + x**28*y**10 + x**28*y**9 + x**28*y**7 - x**28 + x**27*y**26 - x**27*y**19 - x**27*y**17 - x**27*y**16 + x**27*y** 10 + x**27*y**9 + x**27*y**7 - x**27 - y**26 + y**19 + y**17 + y**16 - y**10 - y**9 - y**7 + 1)$ % The symmetric algebra : setring getring m$ setideal(sym,sym(m,{a,b,c})); {y**2*a - z**2*b - x**3*c, x*a - y*b - z*c}$ modequalp(rees,sym); yes$ % Symbolic powers : setring getring m$ setideal(m2,idealpower(m,2)); {x**6*y**2 - 2*x**3*y*z**3 + z**6, x**8 - 2*x**4*y**2*z + y**4*z**2, y**6 - 2*x*y**3*z**2 + x**2*z**4, x**7*y - x**3*y**3*z - x**4*z**3 + y**2*z**4, x**3*y**4 - x**4*y*z**2 - y**3*z**3 + x*z**5, x**4*y**3 - y**5*z - x**5*z**2 + x*y**2*z**3}$ % Let's compute a second symbolic power : setideal(m3,symbolic_power(m,2)); {x**6*y**2 - 2*x**3*y*z**3 + z**6, x**8 - 2*x**4*y**2*z + y**4*z**2, y**6 - 2*x*y**3*z**2 + x**2*z**4, x**2*y**5 + x**7*z - 3*x**3*y**2*z**2 + y*z**5, x**7*y - x**3*y**3*z - x**4*z**3 + y**2*z**4, x**3*y**4 - x**4*y*z**2 - y**3*z**3 + x*z**5, x**4*y**3 - y**5*z - x**5*z**2 + x*y**2*z**3}$ % It is different from the ordinary second power. % Hence m2 has a trivial component. modequalp(m2,m3); no$ % Test x for non zero divisor property : nzdp(x,m2); no$ nzdp(x,m3); yes$ % Here is the primary decomposition : pd:=primarydecomposition m2; pd := {{{x**8 - 2*x**4*y**2*z + y**4*z**2, x**6*y**2 - 2*x**3*y*z**3 + z**6, x**2*z**4 - 2*x*y**3*z**2 + y**6, x**7*z - 3*x**3*y**2*z**2 + x**2*y**5 + y*z**5, x**7*y - x**4*z**3 - x**3*y**3*z + y**2*z**4, - x**4*y*z**2 + x**3*y**4 + x*z**5 - y**3*z**3, - x**5*z**2 + x**4*y**3 + x*y**2*z**3 - y**5*z}, { - x*z**2 + y**3, x**4 - y**2*z, x**3*y - z**3}}, {{z**2, x**6*y**2, y**6, x**2*y**5*z, x**3*y**4, x**4*(x**4 - 2*y**2*z), x**3*y*(x**4 - y**2*z), y**3*(x**4 - y**2*z)}, {x,z,y}}}$ % Compare the result with m2 : setideal(m4,matintersect(first first pd, first second pd)); {y**6 - 2*x*y**3*z**2 + x**2*z**4, x**6*y**2 - 2*x**3*y*z**3 + z**6, x**8 - 2*x**4*y**2*z + y**4*z**2, x**2*y**5*z + x**7*z**2 - 3*x**3*y**2*z**3 + y*z**6, x**4*y**3 - y**5*z - x**5*z**2 + x*y**2*z**3, x**3*y**4 - x**4*y*z**2 - y**3*z**3 + x*z**5, x**7*y - x**3*y**3*z - x**4*z**3 + y**2*z**4}$ modequalp(m2,m4); yes$ % Compare the result with m3 : setideal(m4,first first pd)$ modequalp(m3,m4); yes$ % The trivial component can also be removed with a stable % quotient computation : setideal(m5,matstabquot(m2,vars))$ modequalp(m3,m5); yes$ % Example 3 : The Macaulay curve. setideal(m,proj_monomial_curve({0,1,3,4},{w,x,y,z})); {x**3 - w**2*y, w*y**2 - x**2*z, y**3 - x*z**2, x*y - w*z}$ vars:=first getring(); vars := {w,x,y,z}$ gbasis m; {x**3 - w**2*y, w*y**2 - x**2*z, y**3 - x*z**2, x*y - w*z}$ % Test whether m is prime : isprime m; yes$ % A resolution of m : resolve m; { mat((x**3 - w**2*y),(w*y**2 - x**2*z),(y**3 - x*z**2),(x*y - w*z))$ , mat((y,w,0, - x**2),(z,x,0, - w*y),(0, - y,w, - x*z),(0, - z,x, - y**2))$ , mat((z, - y, - x,w))$ , mat((0))$ }$ % m has depth = 1 as can be seen from the gradedbettinumbers m; {{0},{2,3,3,3},{4,4,4,4},{5}}$ % Another way to see the non perfectness of m : hilbertseries m; ( - w**3 + 2*w**2 + 2*w + 1)/(w**2 - 2*w + 1)$ % Just a third approach. Divide out a parameter system : ps:=for i:=1:2 collect random_linear_form(vars,1000); ps := {927*w + 880*x + 292*y + 9*z, - 819*w + 224*x - 572*y - 205*z}$ setideal(m1,matsum(m,ps))$ % dim should be zero and degree > degree m = 4. % A Gbasis for m1 is computed automatically. dim m1; 0$ degree m1; 5$ % The projections of m on the coord. hyperplanes. for each x in vars collect eliminate(m,{x}); {{ - x*z**2 + y**3}, { - w*z**3 + y**4}, { - w**3*z + x**4}, { - w**2*y + x**3}}$ % Example 4 : Two submodules of S^4. % Get the stored result of the earlier computation. r:=resolve m$ % See whether cali!=degrees contains a relict from earlier % computations. getdegrees(); {}$ % Introduce the 2nd and 3rd syzygy module as new modules. % Both are submodules in S^4. setmodule(m1,second r)$ setmodule(m2,third r)$ % The second is already a gbasis. setgbasis m2; mat((z, - y, - x,w))$ getleadterms m1; mat((0,x**3,0,0),(0,0,x**3,0),(0,0,w**2*y,0),(0,0,w*y**2,0),(0,0,w*x,0),(0,0,0, x**2),(0,0,0,w*y),(0,0,0,x*z),(0,0,0,y**2))$ getleadterms m2; mat((0,0,0,w))$ % Since rk(F/M)=rk(F/in(M)), they have ranks 1 resp. 3. dim m1; 4$ indepvarsets m1; {{w,x,y,z}}$ % Its intersection is zero : matintersect(m1,m2); mat((0,0,0,0))$ % Its sum : setmodule(m3,matsum(m1,m2)); mat(( - y, - w,0,x**2),(0,y, - w,x*z),(0,z, - x,y**2),(z, - y, - x,w),( - y*z - z,y**2 - x,x*y,0))$ dim m3; 3$ % Hence it has a nontrivial annihilator : annihilator m3; {w**2*y*z + w**2*z - w*x*y + w*y**3 - x**3*z - x**2*y*z - x**2*z**2 + x*y**3}$ % One can compute isolated primes and primary decomposition also for % modules. Let's do it, although being trivial here: isolatedprimes m3; {{w**2*y*z + w**2*z - w*x*y + w*y**3 - x**3*z - x**2*y*z - x**2*z**2 + x*y**3}} $ primarydecomposition m3; {{ mat((z*( - w*x*z - w*y*z + w*y - w*z + x**2*z + x*y*z - y**3 + y**2*z),w**2*y + w**2*z + w*x*y*z + w*x*z**2 - w*x*z + w*y*z**2 - x**2*z**2 + x*y**2*z , - w**3,0),(z*( - w*y - w - x*z + y**2), - w*x + w*y*z - x**2*z + x*y**2,w**2* y,0),( - x*z**2,w*y + x*y*z + x*z**2 - y**3,w*( - w + y**2),0),( - x*z**2,y*(w + x*z), - w**2 + x**2*z,0),( - y*z, - (w*z + x*y),w*x,0),( - w*y**2 + x**2*z, - w**2*y + x**3,0,0),(w*y*z - w*y + w*z - x**2*z + y**3 - y**2*z, - w**2 + w*x - w*y*z + x**2* y + x**2*z - x*y**2,0,0),( - w*y*z - w*z - y**3 + y**2*z, - w*x + w*y*z - x**2* z + x*y**2,x**3,0),(z*( - w*y - w + y**2), - w*x + w*y**2 + w*y*z + x*y**2,0,0) ,( - z*(y + 1), - x + y**2,x*y,0),(z, - y, - x,w),(w**2*y*z + w**2*z - w*x*y + w*y**3 - x**3*z - x**2*y*z - x**2*z**2 + x*y **3,0,0,0),( - y, - w,0,x**2),(0,y, - w,x*z),(0,z, - x,y**2))$ , {w**2*y*z + w**2*z - w*x*y + w*y**3 - x**3*z - x**2*y*z - x**2*z**2 + x*y**3}}} $ % To get a meaningful Hilbert series make m1 homogeneous : setdegrees {1,x,x,x}; {1,x,x,x}$ % Reevaluate m1 with the new column degrees. setmodule(m1,m1)$ hilbertseries m1; (w**7 - 5*w**6 + 8*w**5 - 2*w**4 - 5*w**3 + 3*w + 1)/(w**4 - 4*w**3 + 6*w**2 - 4*w + 1)$ % Example 5 : From the MACAULAY manual (D.Bayer, M.Stillman). % An elliptic curve on the Veronese in P^5. rvars:={x,y,z}$ svars:={a,b,c,d,e,f}$ r:=setring(rvars,degreeorder rvars,revlex)$ s:=setring(svars,{for each x in svars collect 2},revlex)$ map:={s,r,{a=x^2,b=x*y,c=x*z,d=y^2,e=y*z,f=z^2}}; map := {{{a, b, c, d, e, f}, {{2,2,2,2,2,2}}, revlex, {1,1,1,1,1,1}}, {{x,y,z}, {{1,1,1}}, revlex, {1,1,1}}, {a=x**2, b=x*y, c=x*z, d=y**2, e=y*z, f=z**2}}$ preimage({y^2z-x^3-x*z^2},map); { - a*d + b**2, - a*e + b*c, - b*e + c*d, - a*f + c**2, - b*f + c*e, - d*f + e**2, a**2 + a*f - b*e, a*b + b*f - d*e, a*c + c*f - d*f}$ % Example 6 : The preimage under a rational map. r:=setring({x,y},{},lex)$ s:=setring({t},{},lex)$ map:={r,s,{x=2t/(t^2+1),y=(t^2-1)/(t^2+1)}}; map := {{{x,y},{},lex,{1,1}}, {{t},{},lex,{1}}, {x=(2*t)/(t**2 + 1),y=(t**2 - 1)/(t**2 + 1)}}$ % The preimage of (0) is the equation of the circle : ratpreimage({},map); {x**2 + y**2 - 1}$ % The preimage of the point (t=3/2) : ratpreimage({2t-3},map); {13*x - 12,13*y - 5}$ % Example 7 : A zerodimensional ideal. setring({x,y,z},{},lex)$ setideal(n,{x**2 + y + z - 3,x + y**2 + z - 3,x + y + z**2 - 3}); {x**2 + y + z - 3,x + y**2 + z - 3,x + y + z**2 - 3}$ % The groebner algorithm with factorization : groebfactor n; {{y - 1,z - 1,x - 1}, {y + 3,z + 3,x + 3}, {y - z,z**2 - 2,x + z - 1}, {z**2 - 2,x - z,y + z - 1}, {y + z - 1,z**2 - 2*z - 1,x + z - 1}}$ % Change the term order and reevaluate n : setring({x,y,z},{{1,1,1}},revlex)$ setideal(n,n); {x**2 + y + z - 3,y**2 + x + z - 3,z**2 + x + y - 3}$ % its primes : zeroprimes n; {{x - z,z**2 - 2,y + z - 1}, {x + z - 1,y + z - 1,z**2 - 2*z - 1}, {z - 1,x - 1,y - 1}, {z + 3,x + 3,y + 3}, {y - z,z**2 - 2,x + z - 1}}$ % a vector space basis of S/n : getkbase n; {1,x,x*y,x*y*z,x*z,y,y*z,z}$ % Example 8 : A modular computation. Since REDUCE has no multivariate % factorizer, factorprimes has to be turned off ! on modular$ off factorprimes$ setmod 181; 1$ setideal(n1,n); {x**2 + y + z + 178,y**2 + x + z + 178,z**2 + x + y + 178}$ zeroprimes n1; {{y + 180*z,z**2 + 179,x + z + 180}, {x + z + 180,y + z + 180,z**2 + 179*z + 180}, {x + 180*z,z**2 + 179,y + z + 180}, {z + 180,x + 180,y + 180}, {z + 3,x + 3,y + 3}}$ setmod 7; 181$ setideal(n1,n); {x**2 + y + z + 4,y**2 + x + z + 4,z**2 + x + y + 4}$ zeroprimes n1; {{z + 6,x + 6,y + 6}, {x + 4,z + 4,y + 2}, {x + 4,z + 2,y + 4}, {z + 4,x + 2,y + 4}, {x + 3,z + 3,y + 3}}$ % Hence some of the primes glue together mod 7. zeroprimarydecomposition n1; {{{z + 6,x + 6,y + 6}, {z + 6,x + 6,y + 6}}, {{z + 4,y + 2,x + 4}, {x + 4,z + 4,y + 2}}, {{z + 2,y + 4,x + 4}, {x + 4,z + 2,y + 4}}, {{z + 4,x + 2,y + 4}, {z + 4,x + 2,y + 4}}, {{x**2 + y + z + 4, x + y**2 + z + 4, x + y + z**2 + 4, 3*(x + 5*y*z + 2*y + 2*z + 5), x*z + 6*x + 3*y + 6*z + 1, x*y + 6*x + 6*y + 3*z + 1}, {x + 3,z + 3,y + 3}}}$ off modular$ on factorprimes$ % Example 9 : Independent sets once more. n:=10$ vars:=for i:=1:(2*n) collect mkid(x,i)$ setring(vars,{},lex)$ setideal(m,for j:=0:n collect for i:=(j+1):(j+n) product mkid(x,i)); {x1*x2*x3*x4*x5*x6*x7*x8*x9*x10, x2*x3*x4*x5*x6*x7*x8*x9*x10*x11, x3*x4*x5*x6*x7*x8*x9*x10*x11*x12, x4*x5*x6*x7*x8*x9*x10*x11*x12*x13, x5*x6*x7*x8*x9*x10*x11*x12*x13*x14, x6*x7*x8*x9*x10*x11*x12*x13*x14*x15, x7*x8*x9*x10*x11*x12*x13*x14*x15*x16, x8*x9*x10*x11*x12*x13*x14*x15*x16*x17, x9*x10*x11*x12*x13*x14*x15*x16*x17*x18, x10*x11*x12*x13*x14*x15*x16*x17*x18*x19, x11*x12*x13*x14*x15*x16*x17*x18*x19*x20}$ setgbasis m$ indepvarsets m; {{x2,x3,x4,x5,x6,x7,x8,x9,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x3,x4,x5,x6,x7,x8,x9,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x4,x5,x6,x7,x8,x9,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x5,x6,x7,x8,x9,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x6,x7,x8,x9,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x7,x8,x9,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x8,x9,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x9,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x9,x12,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x3,x4,x5,x6,x7,x8,x9,x10,x11,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x4,x5,x6,x7,x8,x9,x10,x11,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x5,x6,x7,x8,x9,x10,x11,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x6,x7,x8,x9,x10,x11,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x7,x8,x9,x10,x11,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x8,x9,x10,x11,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x9,x10,x11,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x10,x11,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x13,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x4,x5,x6,x7,x8,x9,x10,x11,x12,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x5,x6,x7,x8,x9,x10,x11,x12,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x6,x7,x8,x9,x10,x11,x12,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x7,x8,x9,x10,x11,x12,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x8,x9,x10,x11,x12,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x9,x10,x11,x12,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x10,x11,x12,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x12,x14,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x5,x6,x7,x8,x9,x10,x11,x12,x13,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x6,x7,x8,x9,x10,x11,x12,x13,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x7,x8,x9,x10,x11,x12,x13,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x8,x9,x10,x11,x12,x13,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x9,x10,x11,x12,x13,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x10,x11,x12,x13,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x12,x13,x15,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x6,x7,x8,x9,x10,x11,x12,x13,x14,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x7,x8,x9,x10,x11,x12,x13,x14,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x8,x9,x10,x11,x12,x13,x14,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x9,x10,x11,x12,x13,x14,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x10,x11,x12,x13,x14,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x12,x13,x14,x16,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x7,x8,x9,x10,x11,x12,x13,x14,x15,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x8,x9,x10,x11,x12,x13,x14,x15,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x9,x10,x11,x12,x13,x14,x15,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x10,x11,x12,x13,x14,x15,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x12,x13,x14,x15,x17,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x8,x9,x10,x11,x12,x13,x14,x15,x16,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x9,x10,x11,x12,x13,x14,x15,x16,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x10,x11,x12,x13,x14,x15,x16,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x12,x13,x14,x15,x16,x18,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x9,x10,x11,x12,x13,x14,x15,x16,x17,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x10,x11,x12,x13,x14,x15,x16,x17,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x12,x13,x14,x15,x16,x17,x19,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x10,x11,x12,x13,x14,x15,x16,x17,x18,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x12,x13,x14,x15,x16,x17,x18,x20}, {x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x12,x13,x14,x15,x16,x17,x18,x19}}$ dim m; 18$ degree m; 55$ comment #################################### ### ### ### Local Standard Bases ### ### ### #################################### end comment; % Example 10 : An example from [ Alonso, Mora, Raimondo ] vars := {z,x,y}$ r:=setring(vars,{},lex)$ setideal(m,{x^3+(x^2-y^2)*z+z^4,y^3+(x^2-y^2)*z-z^4}); {z**4 + z*x**2 - z*y**2 + x**3, - z**4 + z*x**2 - z*y**2 + y**3}$ dim m; 1$ degree m; 12$ % 2 = codim m is the codimension of the curve m. The defining % equations of the singular locus with their nilpotent structure : singular_locus(m,2); {x**3 - y**3 + 2*z**4, x**3 + 2*x**2*z + y**3 - 2*y**2*z, y*(8*x**3 + 3*x**2*y - 11*y**3 + 12*y*z**3), y*(x**3 + 3*x**2*y + 2*x*y*z + y**3 - 2*y**2*z), 3*x**5 + 3*x**4*y + 22*x**4 + 18*x**3*y**2 + 16*x**3*y + 21*x**2*y**3 + 3*x*y **4 - 16*x*y**3 + 18*y**5 - 42*y**4*z - 22*y**4 + 24*y**3*z**2}$ groebfactor ws; {{y,x,z},{81*x + 256,27*z - 64,81*y - 256}}$ % Hence this curve has two singular points : % (x=y=z=0) and (y=-x=256/81,z=64/27) % Let's find the brances of the curve through the origin. % The first critical tropism is (-1,-1,-1). off noetherian$ setring(vars,{{-1,-1,-1}},lex)$ setideal(m,m); {z*x**2 - z*y**2 + x**3 + z**4, z*x**2 - z*y**2 + y**3 - z**4}$ % Let's first test two different approaches, not fully % integrated into the algebraic interface : setideal(m1,homstbasis m); {x**3 - y**3 + 2*z**4, z*x**2 - z*y**2 + y**3 - z**4, z*x*y**2 - z*y**3 - x*y**3 + 2*z**5 + z**4*x, x**2*y**3 + x*y**4 + y**5 - 2*z**5*x - 2*z**5*y - z**4*x**2 - z**4*x*y - z**4* y**2, 6*z*y**5 + 2*x*y**5 - 2*y**6 - 4*z**6*x - 4*z**6*y - 2*z**5*x*y - 8*z**5*y**2 + z**4*x**3 - 2*z**4*x*y**2 + 3*z**4*y**3}$ setideal(m2,lazystbasis m); {x**3 - y**3 + 2*z**4, z*x**2 - z*y**2 + y**3 - z**4, z*x*y**2 - z*y**3 - x*y**3 + 2*z**5 + z**4*x, x**2*y**3 + x*y**4 + y**5 - 2*z**5*x - 2*z**5*y - z**4*x**2 - z**4*x*y - z**4* y**2, 3*z*y**5 + x*y**5 - y**6 - 2*z**6*x - 2*z**6*y - z**5*x**2 - z**5*x*y - 3*z**5 *y**2 - z**4*x*y**2 + z**4*y**3}$ setgbasis m1$ setgbasis m2$ modequalp(m1,m2); yes$ gbasis m; {x**3 - y**3 + 2*z**4, z*x**2 - z*y**2 + y**3 - z**4, z*x*y**2 - z*y**3 - x*y**3 + 2*z**5 + z**4*x, x**2*y**3 + x*y**4 + y**5 - 2*z**5*x - 2*z**5*y - z**4*x**2 - z**4*x*y - z**4* y**2, 3*z*y**5 + x*y**5 - y**6 - 2*z**6*x - 2*z**6*y - z**5*x**2 - z**5*x*y - 3*z**5 *y**2 - z**4*x*y**2 + z**4*y**3}$ modequalp(m,m1); yes$ dim m; 1$ degree m; 9$ % Find the tangent directions not in z-direction : tangentcone m; {x**3 - y**3, z*x**2 - z*y**2 + y**3, z*x*y**2 - z*y**3 - x*y**3, x**2*y**3 + x*y**4 + y**5, 3*z*y**5 + x*y**5 - y**6}$ setideal(n,sub(z=1,ws)); {x**3 - y**3, x**2 - y**2 + y**3, x*y**2 - y**3 - x*y**3, x**2*y**3 + x*y**4 + y**5, 3*y**5 + x*y**5 - y**6}$ setring r$ on noetherian$ setideal(n,n)$ degree n; 9$ % The points of n outside the origin. matstabquot(n,{x,y}); {y**2 - 3*y + 3,x - y + 3}$ % Hence there are two branches x=z'*(a-3+x'),y=z'*(a+y'),z=z' % with the algebraic number a : a^2-3a+3=0 % and the new equations for (z',x',y') : setrules {a^2=>3a-3}; {a**2 => 3*a - 3}$ sub(x=z*(a-3+x),y=z*(a+y),m); {z**3*(a**3 + 3*a**2*x - 9*a**2 + 3*a*x**2 - 16*a*x - 2*a*y + 21*a + x**3 - 8*x **2 + 21*x - y**2 + z - 18), z**3*(a**3 + 3*a**2*y + 2*a*x + 3*a*y**2 - 2*a*y - 6*a + x**2 - 6*x + y**3 - y **2 - z + 9)}$ setideal(m1,matqquot(ws,z)); {x**3 + (3*a - 7)*x**2 - (5*a - 6)*x + y**3 + (3*a - 2)*y**2 + (5*a - 9)*y, z - x**2 - (2*a - 6)*x - y**3 - (3*a - 1)*y**2 - (7*a - 9)*y}$ % This defines a loc. smooth system at the origin, since the % jacobian at the origin of the gbasis is nonsingular : off noetherian$ setring getring m; {{z,x,y},{{-1,-1,-1}},lex,{1,1,1}}$ setideal(m1,m1); { - (5*a - 6)*x + (5*a - 9)*y + (3*a - 7)*x**2 + (3*a - 2)*y**2 + x**3 + y**3, z - (2*a - 6)*x - (7*a - 9)*y - x**2 - (3*a - 1)*y**2 - y**3}$ gbasis m1; {(5*a - 6)*x - (5*a - 9)*y - (3*a - 7)*x**2 - (3*a - 2)*y**2 - x**3 - y**3, (5*a - 6)*z + 27*y + (9*a - 18)*x**2 - (18*a - 45)*y**2 - (2*a - 6)*x**3 - (7* a - 12)*y**3}$ % clear the rules previously set. setrules {}; {}$ % Example 11 : The standard basis of another example. % Comparing different approaches. vars:={x,y}$ setring(vars,localorder vars,lex); {{x,y},{{-1,-1}},lex,{1,1}}$ ff:=x^5+y^11+(x+x^3)*y^9; ff := x**5 + x**3*y**9 + x*y**9 + y**11$ setideal(p1,mat2list matjac({ff},vars)); {5*x**4 + y**9 + 3*x**2*y**9, 9*x*y**8 + 11*y**10 + 9*x**3*y**8}$ gbasis p1; {5*x**4 + y**9 + 3*x**2*y**9, 9*x*y**8 + 11*y**10 + 9*x**3*y**8, 73205*y**16 + 6561*y**17 - 32805*x**10*y**8 + 294030*x**6*y**12 - 292820*x**2* y**16 + 120285*x**9*y**10 - 239580*x**5*y**14 + 19683*x**2*y**17}$ gbtestversion 2$ setideal(p2,p1); {5*x**4 + y**9 + 3*x**2*y**9, 9*x*y**8 + 11*y**10 + 9*x**3*y**8}$ gbasis p2; {5*x**4 + y**9 + 3*x**2*y**9, 9*x*y**8 + 11*y**10 + 9*x**3*y**8, 73205*y**16 + 6561*y**17 - 32805*x**10*y**8 + 294030*x**6*y**12 - 292820*x**2* y**16 + 120285*x**9*y**10 - 239580*x**5*y**14 + 19683*x**2*y**17}$ gbtestversion 3$ setideal(p3,p1); {5*x**4 + y**9 + 3*x**2*y**9, 9*x*y**8 + 11*y**10 + 9*x**3*y**8}$ gbasis p3; {5*x**4 + y**9 + 3*x**2*y**9, 9*x*y**8 + 11*y**10 + 9*x**3*y**8, 73205*y**16 + 6561*y**17 - 32805*x**10*y**8 + 294030*x**6*y**12 - 292820*x**2* y**16 + 120285*x**9*y**10 - 239580*x**5*y**14 + 19683*x**2*y**17}$ gbtestversion 1$ modequalp(p1,p2); yes$ modequalp(p1,p3); yes$ dim p1; 0$ degree p1; 40$ % Example 12 : A local intersection wrt. a non inflimited term order. setring({x,y,z},{},revlex); {{x,y,z},{},revlex,{1,1,1}}$ m1:=matintersect({x-y^2,y-x^2},{x-z^2,z-x^2},{y-z^2,z-y^2}); m1 := {y*z - x**3*y*z - x*y*z**2 + x**4*y*z**2 - y**2*z**2 + x**3*y**2*z**2 + x *y**2*z**3 - x**4*y**2*z**3, x*z - x**2*y*z - x**2*z**2 - x*y*z**2 + x**3*y*z**2 + x**2*y**2*z**2 + x **2*y*z**3 - x**3*y**2*z**3, x*y - x**2*y**2 - x**2*y*z - x*y**2*z + x**3*y**2*z + x**2*y**3*z + x**2 *y**2*z**2 - x**3*y**3*z**2}$ % Delete polynomial units post factum : deleteunits ws; {y*z,x*z,x*y}$ % Detecting polynomial units early : on detectunits; m1:=matintersect({x-y^2,y-x^2},{x-z^2,z-x^2},{y-z^2,z-y^2}); m1 := {y*z,x*z,x*y}$ off detectunits; comment #################################### ### ### ### More Advanced Computations ### ### ### #################################### end comment; % Return to a noetherian term order: vars:={x,y,z}$ setring(vars,degreeorder vars,revlex); {{x,y,z},{{1,1,1}},revlex,{1,1,1}}$ on noetherian; % Example 13 : Use of "mod". % Polynomials modulo ideals : setideal(m,{2x^2+y+5,3y^2+z+7,7z^2+x+1}); {2*x**2 + y + 5,3*y**2 + z + 7,7*z**2 + x + 1}$ x^2*y^2*z^2 mod m; ( - x*y*z - 7*x*y - 5*x*z - 35*x - y*z - 7*y - 5*z - 35)/42$ % Lists of polynomials modulo ideals : {x^3,y^3,z^3} mod gbasis m; {(x*( - y - 5))/2,(y*( - z - 7))/3,( - z*(x + 1))/7}$ % Matrices modulo modules : mm:=mat((x^4,y^4,z^4)); mm := mat((x**4,y**4,z**4))$ mm1:=tp<< ideal2mat m>>; mm1 := mat((2*x**2 + y + 5,3*y**2 + z + 7,x + 7*z**2 + 1))$ mm mod mm1; mat(((y**2 + 10*y + 25)/4,( - 6*x**2*y**2 - 2*x**2*z - 14*x**2 + 4*y**4 + 3*y** 3 + 15*y**2 + y*z + 7*y + 5*z + 35)/4,( - 2*x**3 - 14*x**2*z**2 - 2*x** 2 + x*y + 5*x + 7*y*z**2 + y + 4*z**4 + 35*z**2 + 5)/4))$ % Example 14 : Powersums through elementary symmetric functions. vars:={a,b,c,d,e1,e2,e3,e4}$ setring(vars,{},lex)$ m:=interreduce {a+b+c+d-e1, a*b+a*c+a*d+b*c+b*d+c*d-e2, a*b*c+a*b*d+a*c*d+b*c*d-e3, a*b*c*d-e4}; m := {d**4 - d**3*e1 + d**2*e2 - d*e3 + e4, a + b + c + d - e1, c**3 + c**2*d - c**2*e1 + c*d**2 - c*d*e1 + c*e2 + d**3 - d**2*e1 + d*e2 - e3, b**2 + b*c + b*d - b*e1 + c**2 + c*d - c*e1 + d**2 - d*e1 + e2}$ for n:=1:5 collect a^n+b^n+c^n+d^n mod m; {e1, e1**2 - 2*e2, e1**3 - 3*e1*e2 + 3*e3, e1**4 - 4*e1**2*e2 + 4*e1*e3 + 2*e2**2 - 4*e4, e1**5 - 5*e1**3*e2 + 5*e1**2*e3 + 5*e1*e2**2 - 5*e1*e4 - 5*e2*e3}$ % Example 15 : The setrules mechanism. setring({x,y,z},{},lex)$ setrules {aa^3=>aa+1}; {aa**3 => aa + 1}$ setideal(m,{x^2+y+z-aa,x+y^2+z-aa,x+y+z^2-aa}); {x**2 + y + z - aa,x + y**2 + z - aa,x + y + z**2 - aa}$ gbasis m; {y**2 - y - z**2 + z, x + y + z**2 - aa, 2*y*z**2 - (2*aa - 2)*y + z**4 - (2*aa - 1)*z**2 + (aa**2 - aa), z**6 - (3*aa + 1)*z**4 + 4*z**3 + (3*aa**2 - 2*aa - 2)*z**2 - (4*aa - 4)*z + ( 3*aa**2 - 3*aa - 1)}$ % Clear the rules previously set. setrules {}; {}$ % Example 16 : The same example with advanced coefficient domains. load_package arnum; defpoly aa^3-aa-1; setideal(m,{x^2+y+z-aa,x+y^2+z-aa,x+y+z^2-aa}); {x**2 + y + z - aa,x + y**2 + z - aa,x + y + z**2 - aa}$ gbasis m; {y**2 - y - z**2 + z, x + y + z**2 - aa, y*z**2 - (aa - 1)*y + 1/2*z**4 - (aa - 1/2)*z**2 + (1/2*aa**2 - 1/2*aa), z**6 - (3*aa + 1)*z**4 + 4*z**3 + (3*aa**2 - 2*aa - 2)*z**2 - (4*aa - 4)*z + ( 3*aa**2 - 3*aa - 1)}$ % The following needs some more time since factorization of % arnum's is not so easy : groebfactor m; {{y - (aa**2 - aa - 1), z - (aa**2 - aa - 1), x + (aa**2 - aa - 2)}, {y + (aa**2 - aa - 1), z + (aa**2 - aa - 1), x - (aa**2 - aa)}, {y - z,x - z,z**2 + 2*z - aa}, {z - (aa**2 - aa), y + (aa**2 - aa - 1), x + (aa**2 - aa - 1)}, {z - (aa**2 - aa - 1), y + (aa**2 - aa - 2), x - (aa**2 - aa - 1)}, {z + (aa**2 - aa - 2), y - (aa**2 - aa - 1), x - (aa**2 - aa - 1)}, {z + (aa**2 - aa - 1), y - (aa**2 - aa), x + (aa**2 - aa - 1)}}$ off arnum; off rational; comment #################################### ### ### ### Using Advanced Scripts in ### ### a Complex Example ### ### ### #################################### end comment; % Example 17 : The square of the 2-minors of a symmetric 3x3-matrix. vars:=for i:=1:6 collect mkid(x,i); vars := {x1, x2, x3, x4, x5, x6}$ setring(vars,degreeorder vars,revlex); {{x1,x2,x3,x4,x5,x6},{{1,1,1,1,1,1}},revlex,{1,1,1,1,1,1}}$ % Generating the ideal : mm:=mat((x1,x2,x3),(x2,x4,x5),(x3,x5,x6)); mm := mat((x1,x2,x3),(x2,x4,x5),(x3,x5,x6))$ m:=ideal_of_minors(mm,2); m := { - x1*x4 + x2**2, - x1*x5 + x2*x3, - x1*x6 + x3**2, - x2*x5 + x3*x4, - x2*x6 + x3*x5, - x4*x6 + x5**2}$ setideal(n,idealpower(m,2)); {x2**4 - 2*x1*x2**2*x4 + x1**2*x4**2, x3**4 - 2*x1*x3**2*x6 + x1**2*x6**2, x5**4 - 2*x4*x5**2*x6 + x4**2*x6**2, x2**2*x3**2 - 2*x1*x2*x3*x5 + x1**2*x5**2, x3**2*x5**2 - 2*x2*x3*x5*x6 + x2**2*x6**2, x2**3*x3 - x1*x2*x3*x4 - x1*x2**2*x5 + x1**2*x4*x5, x2*x3**3 - x1*x3**2*x5 - x1*x2*x3*x6 + x1**2*x5*x6, x2**2*x3*x4 - x1*x3*x4**2 - x2**3*x5 + x1*x2*x4*x5, x2**2*x3*x5 - x1*x3*x4*x5 - x2**3*x6 + x1*x2*x4*x6, x2*x3**2*x5 - x1*x3*x5**2 - x2**2*x3*x6 + x1*x2*x5*x6, x3**3*x5 - x2*x3**2*x6 - x1*x3*x5*x6 + x1*x2*x6**2, x2**2*x5**2 - x1*x4*x5**2 - x2**2*x4*x6 + x1*x4**2*x6, x2*x3*x5**2 - x1*x5**3 - x2*x3*x4*x6 + x1*x4*x5*x6, x3*x4*x5**2 - x2*x5**3 - x3*x4**2*x6 + x2*x4*x5*x6, x3*x5**3 - x3*x4*x5*x6 - x2*x5**2*x6 + x2*x4*x6**2, x3**2*x4**2 - 2*x2*x3*x4*x5 + x1*x4*x5**2 + x2**2*x4*x6 - x1*x4**2*x6, x2*x3**2*x4 - 2*x1*x3*x4*x5 + x1*x2*x5**2 - x2**3*x6 + x1*x2*x4*x6, x3**3*x4 - x1*x3*x5**2 - x2**2*x3*x6 - x1*x3*x4*x6 + 2*x1*x2*x5*x6, x3**2*x4*x5 - x1*x5**3 - 2*x2*x3*x4*x6 + x2**2*x5*x6 + x1*x4*x5*x6, x3**2*x4*x6 - 2*x2*x3*x5*x6 + x1*x5**2*x6 + x2**2*x6**2 - x1*x4*x6**2, x1*x3**2*x4 - 2*x1*x2*x3*x5 + x1**2*x5**2 + x1*x2**2*x6 - x1**2*x4*x6}$ % The ideal itself : gbasis n; {x2**4 - 2*x1*x2**2*x4 + x1**2*x4**2, x3**4 - 2*x1*x3**2*x6 + x1**2*x6**2, x5**4 - 2*x4*x5**2*x6 + x4**2*x6**2, x2**2*x3**2 - 2*x1*x2*x3*x5 + x1**2*x5**2, x3**2*x5**2 - 2*x2*x3*x5*x6 + x2**2*x6**2, x2**3*x3 - x1*x2*x3*x4 - x1*x2**2*x5 + x1**2*x4*x5, x2*x3**3 - x1*x3**2*x5 - x1*x2*x3*x6 + x1**2*x5*x6, x2**2*x3*x4 - x1*x3*x4**2 - x2**3*x5 + x1*x2*x4*x5, x2**2*x3*x5 - x1*x3*x4*x5 - x2**3*x6 + x1*x2*x4*x6, x2*x3**2*x5 - x1*x3*x5**2 - x2**2*x3*x6 + x1*x2*x5*x6, x3**3*x5 - x2*x3**2*x6 - x1*x3*x5*x6 + x1*x2*x6**2, x2**2*x5**2 - x1*x4*x5**2 - x2**2*x4*x6 + x1*x4**2*x6, x2*x3*x5**2 - x1*x5**3 - x2*x3*x4*x6 + x1*x4*x5*x6, x3*x4*x5**2 - x2*x5**3 - x3*x4**2*x6 + x2*x4*x5*x6, x3*x5**3 - x3*x4*x5*x6 - x2*x5**2*x6 + x2*x4*x6**2, x3**2*x4**2 - 2*x2*x3*x4*x5 + x1*x4*x5**2 + x2**2*x4*x6 - x1*x4**2*x6, x2*x3**2*x4 - 2*x1*x3*x4*x5 + x1*x2*x5**2 - x2**3*x6 + x1*x2*x4*x6, x3**3*x4 - x1*x3*x5**2 - x2**2*x3*x6 - x1*x3*x4*x6 + 2*x1*x2*x5*x6, x3**2*x4*x5 - x1*x5**3 - 2*x2*x3*x4*x6 + x2**2*x5*x6 + x1*x4*x5*x6, x3**2*x4*x6 - 2*x2*x3*x5*x6 + x1*x5**2*x6 + x2**2*x6**2 - x1*x4*x6**2, x1*x3**2*x4 - 2*x1*x2*x3*x5 + x1**2*x5**2 + x1*x2**2*x6 - x1**2*x4*x6}$ length n; 21$ dim n; 3$ degree n; 16$ % Its radical. radical n; { - x1*x5 + x2*x3, - x2*x5 + x3*x4, - x2*x6 + x3*x5, - x1*x4 + x2**2, - x1*x6 + x3**2, - x4*x6 + x5**2}$ % Its unmixed radical. unmixedradical n; { - x1*x5 + x2*x3, x2*x5 - x3*x4, - x2*x6 + x3*x5, - x1*x4 + x2**2, - x1*x6 + x3**2, - x4*x6 + x5**2}$ % Its equidimensional hull : n1:=eqhull n; n1 := {x1**2*x4**2 - 2*x1*x2**2*x4 + x2**4, x1**2*x6**2 - 2*x1*x3**2*x6 + x3**4, x4**2*x6**2 - 2*x4*x5**2*x6 + x5**4, x1**2*x5**2 - 2*x1*x2*x3*x5 + x2**2*x3**2, x2**2*x6**2 - 2*x2*x3*x5*x6 + x3**2*x5**2, x1**2*x4*x5 - x1*x2**2*x5 - x1*x2*x3*x4 + x2**3*x3, x1**2*x5*x6 - x1*x2*x3*x6 - x1*x3**2*x5 + x2*x3**3, x1*x2*x4*x5 - x1*x3*x4**2 - x2**3*x5 + x2**2*x3*x4, x1*x2*x4*x6 - x1*x3*x4*x5 - x2**3*x6 + x2**2*x3*x5, x1*x2*x5*x6 - x1*x3*x5**2 - x2**2*x3*x6 + x2*x3**2*x5, x1*x2*x6**2 - x1*x3*x5*x6 - x2*x3**2*x6 + x3**3*x5, x1*x4**2*x6 - x1*x4*x5**2 - x2**2*x4*x6 + x2**2*x5**2, x1*x4*x5*x6 - x1*x5**3 - x2*x3*x4*x6 + x2*x3*x5**2, x2*x4*x5*x6 - x2*x5**3 - x3*x4**2*x6 + x3*x4*x5**2, x2*x4*x6**2 - x2*x5**2*x6 - x3*x4*x5*x6 + x3*x5**3, - x1*x4*x6 + x1*x5**2 + x2**2*x6 - 2*x2*x3*x5 + x3**2*x4}$ length n1; 16$ setideal(n1,n1)$ submodulep(n,n1); yes$ submodulep(n1,n); no$ % Hence there is an embedded component. Let's find it making % an excursion to symbolic mode. Of course, this can be done % also algebraically. symbolic; nil n:=get('n,'basis); (dpmat 21 0 ((1 ((((0 0 4) 4) . 1) (((0 1 2 0 1) 4) . -2) (((0 2 0 0 2) 4) . 1) ) 3 0 nil) (2 ((((0 0 0 4) 4) . 1) (((0 1 0 2 0 0 1) 4) . -2) (((0 2 0 0 0 0 2) 4) . 1)) 3 0 nil) (3 ((((0 0 0 0 0 4) 4) . 1) (((0 0 0 0 1 2 1) 4) . -2) (((0 0 0 0 2 0 2) 4) . 1)) 3 0 nil) (4 ((((0 0 2 2) 4) . 1) (((0 1 1 1 0 1) 4) . -2) ( ((0 2 0 0 0 2) 4) . 1)) 3 0 nil) (5 ((((0 0 0 2 0 2) 4) . 1) (((0 0 1 1 0 1 1) 4 ) . -2) (((0 0 2 0 0 0 2) 4) . 1)) 3 0 nil) (6 ((((0 0 3 1) 4) . 1) (((0 1 1 1 1 ) 4) . -1) (((0 1 2 0 0 1) 4) . -1) (((0 2 0 0 1 1) 4) . 1)) 4 0 nil) (7 ((((0 0 1 3) 4) . 1) (((0 1 0 2 0 1) 4) . -1) (((0 1 1 1 0 0 1) 4) . -1) (((0 2 0 0 0 1 1) 4) . 1)) 4 0 nil) (8 ((((0 0 2 1 1) 4) . 1) (((0 1 0 1 2) 4) . -1) (((0 0 3 0 0 1) 4) . -1) (((0 1 1 0 1 1) 4) . 1)) 4 0 nil) (9 ((((0 0 2 1 0 1) 4) . 1) ((( 0 1 0 1 1 1) 4) . -1) (((0 0 3 0 0 0 1) 4) . -1) (((0 1 1 0 1 0 1) 4) . 1)) 4 0 nil) (10 ((((0 0 1 2 0 1) 4) . 1) (((0 1 0 1 0 2) 4) . -1) (((0 0 2 1 0 0 1) 4) . -1) (((0 1 1 0 0 1 1) 4) . 1)) 4 0 nil) (11 ((((0 0 0 3 0 1) 4) . 1) (((0 0 1 2 0 0 1) 4) . -1) (((0 1 0 1 0 1 1) 4) . -1) (((0 1 1 0 0 0 2) 4) . 1)) 4 0 nil ) (12 ((((0 0 2 0 0 2) 4) . 1) (((0 1 0 0 1 2) 4) . -1) (((0 0 2 0 1 0 1) 4) . -1) (((0 1 0 0 2 0 1) 4) . 1)) 4 0 nil) (13 ((((0 0 1 1 0 2) 4) . 1) (((0 1 0 0 0 3) 4) . -1) (((0 0 1 1 1 0 1) 4) . -1) (((0 1 0 0 1 1 1) 4) . 1)) 4 0 nil) ( 14 ((((0 0 0 1 1 2) 4) . 1) (((0 0 1 0 0 3) 4) . -1) (((0 0 0 1 2 0 1) 4) . -1) (((0 0 1 0 1 1 1) 4) . 1)) 4 0 nil) (15 ((((0 0 0 1 0 3) 4) . 1) (((0 0 0 1 1 1 1) 4) . -1) (((0 0 1 0 0 2 1) 4) . -1) (((0 0 1 0 1 0 2) 4) . 1)) 4 0 nil) (16 ( (((0 0 0 2 2) 4) . 1) (((0 0 1 1 1 1) 4) . -2) (((0 1 0 0 1 2) 4) . 1) (((0 0 2 0 1 0 1) 4) . 1) (((0 1 0 0 2 0 1) 4) . -1)) 5 0 nil) (17 ((((0 0 1 2 1) 4) . 1 ) (((0 1 0 1 1 1) 4) . -2) (((0 1 1 0 0 2) 4) . 1) (((0 0 3 0 0 0 1) 4) . -1) ( ((0 1 1 0 1 0 1) 4) . 1)) 5 0 nil) (18 ((((0 0 0 3 1) 4) . 1) (((0 1 0 1 0 2) 4 ) . -1) (((0 0 2 1 0 0 1) 4) . -1) (((0 1 0 1 1 0 1) 4) . -1) (((0 1 1 0 0 1 1) 4) . 2)) 5 0 nil) (19 ((((0 0 0 2 1 1) 4) . 1) (((0 1 0 0 0 3) 4) . -1) (((0 0 1 1 1 0 1) 4) . -2) (((0 0 2 0 0 1 1) 4) . 1) (((0 1 0 0 1 1 1) 4) . 1)) 5 0 nil) (20 ((((0 0 0 2 1 0 1) 4) . 1) (((0 0 1 1 0 1 1) 4) . -2) (((0 1 0 0 0 2 1) 4) . 1) (((0 0 2 0 0 0 2) 4) . 1) (((0 1 0 0 1 0 2) 4) . -1)) 5 0 nil) (21 ((((0 1 0 2 1) 4) . 1) (((0 1 1 1 0 1) 4) . -2) (((0 2 0 0 0 2) 4) . 1) (((0 1 2 0 0 0 1) 4) . 1) (((0 2 0 0 1 0 1) 4) . -1)) 5 0 nil)) nil nil) % This needs even more time than the eqhull, of course. u:=primarydecomposition!* n; (((dpmat 16 0 ((1 ((((0 0 2 1 0 1) 4) . 1) (((0 1 0 1 1 1) 4) . -1) (((0 0 3 0 0 0 1) 4) . -1) (((0 1 1 0 1 0 1) 4) . 1)) 4 0 nil) (2 ((((0 0 1 2 0 1) 4) . 1) ( ((0 1 0 1 0 2) 4) . -1) (((0 0 2 1 0 0 1) 4) . -1) (((0 1 1 0 0 1 1) 4) . 1)) 4 0 nil) (3 ((((0 0 1 1 0 2) 4) . 1) (((0 1 0 0 0 3) 4) . -1) (((0 0 1 1 1 0 1) 4 ) . -1) (((0 1 0 0 1 1 1) 4) . 1)) 4 0 nil) (4 ((((0 0 4) 4) . 1) (((0 1 2 0 1) 4) . -2) (((0 2 0 0 2) 4) . 1)) 3 0 nil) (5 ((((0 0 3 1) 4) . 1) (((0 1 1 1 1) 4 ) . -1) (((0 1 2 0 0 1) 4) . -1) (((0 2 0 0 1 1) 4) . 1)) 4 0 nil) (6 ((((0 0 2 2) 4) . 1) (((0 1 1 1 0 1) 4) . -2) (((0 2 0 0 0 2) 4) . 1)) 3 0 nil) (7 ((((0 0 2 1 1) 4) . 1) (((0 1 0 1 2) 4) . -1) (((0 0 3 0 0 1) 4) . -1) (((0 1 1 0 1 1) 4 ) . 1)) 4 0 nil) (8 ((((0 0 2 0 0 2) 4) . 1) (((0 1 0 0 1 2) 4) . -1) (((0 0 2 0 1 0 1) 4) . -1) (((0 1 0 0 2 0 1) 4) . 1)) 4 0 nil) (9 ((((0 0 0 4) 4) . 1) ((( 0 1 0 2 0 0 1) 4) . -2) (((0 2 0 0 0 0 2) 4) . 1)) 3 0 nil) (10 ((((0 0 0 0 0 4 ) 4) . 1) (((0 0 0 0 1 2 1) 4) . -2) (((0 0 0 0 2 0 2) 4) . 1)) 3 0 nil) (11 (( ((0 0 0 2 0 2) 4) . 1) (((0 0 1 1 0 1 1) 4) . -2) (((0 0 2 0 0 0 2) 4) . 1)) 3 0 nil) (12 ((((0 0 1 3) 4) . 1) (((0 1 0 2 0 1) 4) . -1) (((0 1 1 1 0 0 1) 4) . -1 ) (((0 2 0 0 0 1 1) 4) . 1)) 4 0 nil) (13 ((((0 0 0 3 0 1) 4) . 1) (((0 0 1 2 0 0 1) 4) . -1) (((0 1 0 1 0 1 1) 4) . -1) (((0 1 1 0 0 0 2) 4) . 1)) 4 0 nil) ( 14 ((((0 0 0 1 1 2) 4) . 1) (((0 0 1 0 0 3) 4) . -1) (((0 0 0 1 2 0 1) 4) . -1) (((0 0 1 0 1 1 1) 4) . 1)) 4 0 nil) (15 ((((0 0 0 1 0 3) 4) . 1) (((0 0 0 1 1 1 1) 4) . -1) (((0 0 1 0 0 2 1) 4) . -1) (((0 0 1 0 1 0 2) 4) . 1)) 4 0 nil) (16 ( (((0 0 0 2 1) 3) . 1) (((0 0 1 1 0 1) 3) . -2) (((0 1 0 0 0 2) 3) . 1) (((0 0 2 0 0 0 1) 3) . 1) (((0 1 0 0 1 0 1) 3) . -1)) 5 0 nil)) nil t) (dpmat 6 0 ((1 (( ((0 0 0 1 0 1) 2) . 1) (((0 0 1 0 0 0 1) 2) . -1)) 2 0 nil) (2 ((((0 0 0 0 0 2) 2) . 1) (((0 0 0 0 1 0 1) 2) . -1)) 2 0 nil) (3 ((((0 0 0 1 1) 2) . 1) (((0 0 1 0 0 1) 2) . -1)) 2 0 nil) (4 ((((0 0 0 2) 2) . 1) (((0 1 0 0 0 0 1) 2) . -1)) 2 0 nil) (5 ((((0 0 1 1) 2) . 1) (((0 1 0 0 0 1) 2) . -1)) 2 0 nil) (6 ((((0 0 2) 2) . 1) (((0 1 0 0 1) 2) . -1)) 2 0 nil)) nil t)) ((dpmat 18 0 ((1 ((((0 0 1 0 0 3) 4) . 1)) 1 0 nil) (2 ((((0 0 0 0 0 0 1) 1) . 1)) 1 0 nil) (3 ((((0 0 0 4) 4) . 1)) 1 0 nil) (4 ((((0 0 0 0 0 4) 4) . 1)) 1 0 nil) (5 ((((0 0 0 2 0 2) 4) . 1 )) 1 0 nil) (6 ((((0 0 0 3 0 1) 4) . 1)) 1 0 nil) (7 ((((0 0 0 1 0 3) 4) . 1)) 1 0 nil) (8 ((((0 0 0 0 1) 1) . 1)) 1 0 nil) (9 ((((0 0 3 0 0 1) 4) . 1)) 1 0 nil ) (10 ((((0 0 2 1 0 1) 4) . 1)) 1 0 nil) (11 ((((0 0 1 2 0 1) 4) . 1)) 1 0 nil) (12 ((((0 0 2 0 0 2) 4) . 1)) 1 0 nil) (13 ((((0 0 1 1 0 2) 4) . 1)) 1 0 nil) ( 14 ((((0 0 4) 4) . 1)) 1 0 nil) (15 ((((0 0 2 2) 4) . 1)) 1 0 nil) (16 ((((0 1) 1) . 1)) 1 0 nil) (17 ((((0 0 1 3) 4) . 1)) 1 0 nil) (18 ((((0 0 3 1) 4) . 1)) 1 0 nil)) nil t) (dpmat 6 0 ((1 ((((0 0 0 0 0 0 1) 1) . 1)) 1 0 nil) (2 ((((0 0 0 0 1) 1) . 1)) 1 0 nil) (3 ((((0 1) 1) . 1)) 1 0 nil) (4 ((((0 0 1) 1) . 1)) 1 0 nil) (5 ((((0 0 0 1) 1) . 1)) 1 0 nil) (6 ((((0 0 0 0 0 1) 1) . 1)) 1 0 nil)) nil t))) for each x in u collect easydim!* cadr x; (3 0) for each x in u collect degree!* car x; (16 20) % Hence the embedded component is a trivial one. Let's divide % it out by a stable ideal quotient calculation : algebraic; setideal(n2,matstabquot(n,vars)); {x2**4 - 2*x1*x2**2*x4 + x1**2*x4**2, x3**4 - 2*x1*x3**2*x6 + x1**2*x6**2, x5**4 - 2*x4*x5**2*x6 + x4**2*x6**2, x2**2*x3**2 - 2*x1*x2*x3*x5 + x1**2*x5**2, x3**2*x5**2 - 2*x2*x3*x5*x6 + x2**2*x6**2, x2**3*x3 - x1*x2*x3*x4 - x1*x2**2*x5 + x1**2*x4*x5, x2*x3**3 - x1*x3**2*x5 - x1*x2*x3*x6 + x1**2*x5*x6, x2**2*x3*x4 - x1*x3*x4**2 - x2**3*x5 + x1*x2*x4*x5, x2**2*x3*x5 - x1*x3*x4*x5 - x2**3*x6 + x1*x2*x4*x6, x2*x3**2*x5 - x1*x3*x5**2 - x2**2*x3*x6 + x1*x2*x5*x6, x3**3*x5 - x2*x3**2*x6 - x1*x3*x5*x6 + x1*x2*x6**2, x2**2*x5**2 - x1*x4*x5**2 - x2**2*x4*x6 + x1*x4**2*x6, x2*x3*x5**2 - x1*x5**3 - x2*x3*x4*x6 + x1*x4*x5*x6, x3*x4*x5**2 - x2*x5**3 - x3*x4**2*x6 + x2*x4*x5*x6, x3*x5**3 - x3*x4*x5*x6 - x2*x5**2*x6 + x2*x4*x6**2, x3**2*x4 - 2*x2*x3*x5 + x1*x5**2 + x2**2*x6 - x1*x4*x6}$ modequalp(n1,n2); yes$ comment ######################################## ### ### ### Test Examples for New Features ### ### ### ######################################## end comment; % ==> Testing the different zerodimensional solver vars:={x,y,z}$ setring(vars,degreeorder vars,revlex); {{x,y,z},{{1,1,1}},revlex,{1,1,1}}$ setideal(m,{x^3+y+z-3,y^3+x+z-3,z^3+x+y-3}); {x**3 + y + z - 3,y**3 + x + z - 3,z**3 + x + y - 3}$ zerosolve1 m; {{x + y + z**3 - 3, y + z**3 + z - 3, z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8}, {x + y + z**3 - 3, 2*y + z**3 - 3, z**6 - 2*z**4 - 6*z**3 + 4*z**2 + 6*z + 5}, {x + y + z, y**2 + y*z + z**2 - 1, z**3 - z - 3}, {x + z**3 + z - 3, y - z, z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8}, {x - z,y - z,z**2 + z + 3}, {x - 1,y - 1,z - 1}}$ zerosolve2 m; {{x + y + z**3 - 3, y + z**3 + z - 3, z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8}, {x + y + z**3 - 3, 2*y + z**3 - 3, z**6 - 2*z**4 - 6*z**3 + 4*z**2 + 6*z + 5}, {x + y + z, y**2 + y*z + z**2 - 1, z**3 - z - 3}, {x + z**3 + z - 3, y - z, z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8}, {x - z,y - z,z**2 + z + 3}, {x - 1,y - 1,z - 1}}$ setring(vars,{},lex)$ setideal(m,m)$ m1:=gbasis m$ zerosolve m1; {{x - 1,y - 1,z - 1}, {x - z,y - z,z**2 + z + 3}, {x + y + z, y**2 + y*z + z**2 - 1, z**3 - z - 3}, {2*x + z**3 - 3, 2*y + z**3 - 3, z**6 - 2*z**4 - 6*z**3 + 4*z**2 + 6*z + 5}, {x + z**3 + z - 3, y - z, z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8}, {x - z, y + z**3 + z - 3, z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8}}$ zerosolve1 m1; {{x - 1,y - 1,z - 1}, {x - z,y - z,z**2 + z + 3}, {x + y + z, y**2 + y*z + z**2 - 1, z**3 - z - 3}, {x - y, 2*y + z**3 - 3, z**6 - 2*z**4 - 6*z**3 + 4*z**2 + 6*z + 5}, {x + z**3 + z - 3, y - z, z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8}, {x - z, y + z**3 + z - 3, z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8}}$ zerosolve2 m1; {{x - 1,y - 1,z - 1}, {x - z,y - z,z**2 + z + 3}, {x + y + z, y**2 + y*z + z**2 - 1, z**3 - z - 3}, {x - y, 2*y + z**3 - 3, z**6 - 2*z**4 - 6*z**3 + 4*z**2 + 6*z + 5}, {x + z**3 + z - 3, y - z, z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8}, {x - z, y + z**3 + z - 3, z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8}}$ % ==> Testing groebfactor, extendedgroebfactor, extendedgroebfactor1 % Gerdt et al. : Seventh order KdV type equation. A1:=-2*L1**2+L1*L2+2*L1*L3-L2**2-7*L5+21*L6$ A2:=7*L7-2*L1*L4+3/7*L1**3$ B1:=L1*(5*L1-3*L2+L3)$ B2:=L1*(2*L6-4*L4)$ B3:=L1*L7/2$ P1:=L1*(L4-L5/2+L6)$ P2:=(2/7*L1**2-L4)*(-10*L1+5*L2-L3)$ P3:=(2/7*L1**2-L4)*(3*L4-L5+L6)$ P4:=A1*(-3*L1+2*L2)+21*A2$ P5:=A1*(2*L4-2*L5)+A2*(-45*L1+15*L2-3*L3)$ P6:=2*A1*L7+A2*(12*L4-3*L5+2*L6)$ P7:=B1*(2*L2-L1)+7*B2$ P8:=B1*L3+7*B2$ P9:=B1*(-2*L4-2*L5)+B2*(2*L2-8*L1)+84*B3$ P10:=B1*(8/3*L5+6*L6)+B2*(11*L1-17/3*L2+5/3*L3)-168*B3$ P11:=15*B1*L7+B2*(5*L4-2*L5)+B3*(-120*L1+30*L2-6*L3)$ P12:=-3*B1*L7+B2*(-L4/2+L5/4-L6/2)+B3*(24*L1-6*L2)$ P13:=3*B2*L7+B3*(40*L4-8*L5+4*L6)$ polys:={P1,P2,P3,P4,P5,P6,P7,P8,P9,P10,P11,P12,P13}; polys := {(l1*(2*l4 - l5 + 2*l6))/2, ( - 20*l1**3 + 10*l1**2*l2 - 2*l1**2*l3 + 70*l1*l4 - 35*l2*l4 + 7*l3* l4)/7, (6*l1**2*l4 - 2*l1**2*l5 + 2*l1**2*l6 - 21*l4**2 + 7*l4*l5 - 7*l4*l6) /7, 15*l1**3 - 7*l1**2*l2 - 6*l1**2*l3 + 5*l1*l2**2 + 4*l1*l2*l3 - 42*l1* l4 + 21*l1*l5 - 63*l1*l6 - 2*l2**3 - 14*l2*l5 + 42*l2*l6 + 147*l7, ( - 135*l1**4 + 45*l1**3*l2 - 9*l1**3*l3 + 602*l1**2*l4 + 28*l1**2*l5 - 196*l1*l2*l4 - 14*l1*l2*l5 + 70*l1*l3*l4 - 28*l1*l3*l5 - 2205*l1* l7 - 14*l2**2*l4 + 14*l2**2*l5 + 735*l2*l7 - 147*l3*l7 - 98*l4*l5 + 294*l4*l6 + 98*l5**2 - 294*l5*l6)/7, (36*l1**3*l4 - 9*l1**3*l5 + 6*l1**3*l6 - 28*l1**2*l7 + 14*l1*l2*l7 + 28*l1*l3*l7 - 168*l1*l4**2 + 42*l1*l4*l5 - 28*l1*l4*l6 - 14*l2**2*l7 + 588*l4*l7 - 245*l5*l7 + 392*l6*l7)/7, l1*( - 5*l1**2 + 13*l1*l2 - l1*l3 - 6*l2**2 + 2*l2*l3 - 28*l4 + 14*l6 ), l1*(5*l1*l3 - 3*l2*l3 + l3**2 - 28*l4 + 14*l6), 2*l1*(11*l1*l4 - 5*l1*l5 - 8*l1*l6 - l2*l4 + 3*l2*l5 + 2*l2*l6 - l3* l4 - l3*l5 + 21*l7), (4*l1*( - 33*l1*l4 + 10*l1*l5 + 39*l1*l6 + 17*l2*l4 - 6*l2*l5 - 22*l2 *l6 - 5*l3*l4 + 2*l3*l5 + 7*l3*l6 - 63*l7))/3, l1*(15*l1*l7 - 30*l2*l7 + 12*l3*l7 - 20*l4**2 + 8*l4*l5 + 10*l4*l6 - 4*l5*l6), (l1*( - 6*l1*l7 + 12*l2*l7 - 6*l3*l7 + 4*l4**2 - 2*l4*l5 + 2*l4*l6 + l5*l6 - 2*l6**2))/2, 4*l1*l7*(2*l4 - l5 + 2*l6)}$ vars:={L7,L6,L5,L4,L3,L2,L1}; vars := {l7, l6, l5, l4, l3, l2, l1}$ clear a1,a2,b1,b2,b3$ off lexefgb; setring(vars,{},lex); {{l7,l6,l5,l4,l3,l2,l1},{},lex,{1,1,1,1,1,1,1}}$ % The factorized Groebner algorithm. groebfactor polys; {{l1,l4,l7,21*l6 - 7*l5 - l2**2}, {l1, l4, 7*l5 - l3*l2 + 5*l2**2, 56*l6 - 5*l3*l2 + 23*l2**2, 588*l7 + 7*l3*l2**2 - 37*l2**3}, {l1, l7, l3 - 5*l2, 14*l6 - 21*l4 - l2**2, 14*l5 - 63*l4 - l2**2}, {l1,l4,l2,l5,l7}, {7*l4 - 2*l1**2, l2 - 2*l1, l3 - 3*l1, 147*l7 - 4*l1**3, 7*l5 - 6*l1**2, 7*l6 - l1**2}, {7*l4 - 2*l1**2, 2*l2 - 7*l1, l3 - 6*l1, 147*l7 - 4*l1**3, 7*l5 - 9*l1**2, 14*l6 - 5*l1**2}, {l1, l3 - 5*l2, 63*l4 + 2*l2**2, 63*l5 + 2*l2**2, 63*l6 - 4*l2**2, 1323*l7 + 10*l2**3}, {l1,l2,l3,l7,l5 - l4,l6 + 2*l4}, {l2 - 3*l1, l3 - 5*l1, 14*l4 - 5*l1**2, 98*l7 - 5*l1**3, 7*l5 - 10*l1**2, 14*l6 - 5*l1**2}}$ % The extended Groebner factorizer, producing triangular sets. extendedgroebfactor polys; {{{98*l7 - 5*l1**3, 14*l6 - 5*l1**2, 7*l5 - 10*l1**2, 14*l4 - 5*l1**2, l3 - 5*l1, l2 - 3*l1}, {}, {l1}}, {{l7,l6 + 2*l4,l5 - l4,l3,l2,l1},{},{l4}}, {{1323*l7 + 10*l2**3, 63*l6 - 4*l2**2, 63*l5 + 2*l2**2, 63*l4 + 2*l2**2, l3 - 5*l2, l1}, {}, {l2}}, {{147*l7 - 4*l1**3, 14*l6 - 5*l1**2, 7*l5 - 9*l1**2, 7*l4 - 2*l1**2, l3 - 6*l1, 2*l2 - 7*l1}, {}, {l1}}, {{147*l7 - 4*l1**3, 7*l6 - l1**2, 7*l5 - 6*l1**2, 7*l4 - 2*l1**2, l3 - 3*l1, l2 - 2*l1}, {}, {l1}}, {{l7,l5,l4,l2,l1},{},{l6,l3}}, {{l7, 14*l6 - (l2**2 + 21*l4), 14*l5 - (l2**2 + 63*l4), l3 - 5*l2, l1}, {}, {l4,l2}}, {{588*l7 - (37*l2**3 - 7*l2**2*l3), 56*l6 + (23*l2**2 - 5*l2*l3), 7*l5 + (5*l2**2 - l2*l3), l4, l1}, {}, {l3,l2}}, {{l7,21*l6 - (l2**2 + 7*l5),l4,l1},{},{l5,l3,l2}}}$ % The extended Groebner factorizer with subproblem removal check. extendedgroebfactor1 polys; {{{l7,21*l6 - (l2**2 + 7*l5),l4,l1},{},{l5,l3,l2}}, {{588*l7 - (37*l2**3 - 7*l2**2*l3), 56*l6 + (23*l2**2 - 5*l2*l3), 7*l5 + (5*l2**2 - l2*l3), l4, l1}, {}, {l3,l2}}, {{l7, 14*l6 - (l2**2 + 21*l4), 14*l5 - (l2**2 + 63*l4), l3 - 5*l2, l1}, {}, {l4,l2}}, {{l7,l5,l4,l2,l1},{},{l6,l3}}, {{147*l7 - 4*l1**3, 7*l6 - l1**2, 7*l5 - 6*l1**2, 7*l4 - 2*l1**2, l3 - 3*l1, l2 - 2*l1}, {}, {l1}}, {{147*l7 - 4*l1**3, 14*l6 - 5*l1**2, 7*l5 - 9*l1**2, 7*l4 - 2*l1**2, l3 - 6*l1, 2*l2 - 7*l1}, {}, {l1}}, {{1323*l7 + 10*l2**3, 63*l6 - 4*l2**2, 63*l5 + 2*l2**2, 63*l4 + 2*l2**2, l3 - 5*l2, l1}, {}, {l2}}, {{l7,l6 + 2*l4,l5 - l4,l3,l2,l1},{},{l4}}, {{98*l7 - 5*l1**3, 14*l6 - 5*l1**2, 7*l5 - 10*l1**2, 14*l4 - 5*l1**2, l3 - 5*l1, l2 - 3*l1}, {}, {l1}}}$ % Gonnet's example (ACM SIGSAM Bulletin 17 (1983), 48 - 49) vars:={a0,a2,a3,a4,a5,b0,b1,b2,b3,b4,b5,c0,c1,c2,c3,c4,c5}; vars := {a0, a2, a3, a4, a5, b0, b1, b2, b3, b4, b5, c0, c1, c2, c3, c4, c5}$ polys:={a4*b4, a5*b1+b5+a4*b3+a3*b4, a2*b2,a5*b5, (a0+1+a4)*b2+a2*(b0+b1+b4)+c2, (a0+1+a4)*(b0+b1+b4)+(a3+a5)*b2+a2*(b3+b5)+c0+c1+c4, (a3+a5)*(b0+b1+b4)+(b3+b5)*(a0+1+a4)+c3+c5-1, (a3+a5)*(b3+b5), a5*(b3+b5)+b5*(a3+a5), b5*(a0+1+2*a4)+a5*(b0+b1+2*b4)+a3*b4+a4*b3+c5, a4*(b0+b1+2*b4)+a2*b5+a5*b2+(a0+1)*b4+c4, a2*b4+a4*b2, a4*b5+a5*b4, 2*a3*b3+a3*b5+a5*b3, c3+b3*(a0+2+a4)+a3*(b0+2*b1+b4)+b5+a5*b1, c1+(a0+2+a4)*b1+a2*b3+a3*b2+(b0+b4), a2*b1+b2, a5*b3+a3*b5, b4+a4*b1}; polys := {a4*b4, a3*b4 + a4*b3 + a5*b1 + b5, a2*b2, a5*b5, a0*b2 + a2*b0 + a2*b1 + a2*b4 + a4*b2 + b2 + c2, a0*b0 + a0*b1 + a0*b4 + a2*b3 + a2*b5 + a3*b2 + a4*b0 + a4*b1 + a4*b4 + a5*b2 + b0 + b1 + b4 + c0 + c1 + c4, a0*b3 + a0*b5 + a3*b0 + a3*b1 + a3*b4 + a4*b3 + a4*b5 + a5*b0 + a5*b1 + a5*b4 + b3 + b5 + c3 + c5 - 1, a3*b3 + a3*b5 + a5*b3 + a5*b5, a3*b5 + a5*b3 + 2*a5*b5, a0*b5 + a3*b4 + a4*b3 + 2*a4*b5 + a5*b0 + a5*b1 + 2*a5*b4 + b5 + c5, a0*b4 + a2*b5 + a4*b0 + a4*b1 + 2*a4*b4 + a5*b2 + b4 + c4, a2*b4 + a4*b2, a4*b5 + a5*b4, 2*a3*b3 + a3*b5 + a5*b3, a0*b3 + a3*b0 + 2*a3*b1 + a3*b4 + a4*b3 + a5*b1 + 2*b3 + b5 + c3, a0*b1 + a2*b3 + a3*b2 + a4*b1 + b0 + 2*b1 + b4 + c1, a2*b1 + b2, a3*b5 + a5*b3, a4*b1 + b4}$ on lexefgb; % Switching back to the default. setring(vars,{},lex); {{a0,a2,a3,a4,a5,b0,b1,b2,b3,b4,b5,c0,c1,c2,c3,c4,c5}, {}, lex, {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}}$ groebfactor polys; {{c5, b5, c2, c4, b2, b4, a4, a2, a5, b3, a3*b1 + 1, b0 - b1*c3 + 2*b1, a0 - a3*c1 + c3, b1*c3**2 - 2*b1*c3 + b1 - c0 + c1*c3 - 2*c1, a3*c0 - a3*c1*c3 + 2*a3*c1 + c3**2 - 2*c3 + 1}, {c5, c4, b5, b4, a4, b2, a5, a3, b1, b3 + 1, a0 - c3 + 2, b0*c3 - 2*b0 + c0, a2 - b0 - c1, b0**2 + b0*c1 + c2, b0*c0 + c0*c1 - c2*c3 + 2*c2, c0**2 - c0*c1*c3 + 2*c0*c1 + c2*c3**2 - 4*c2*c3 + 4*c2}, {c5, b5, c2, c4, b2, b4, a4, a2, a5, a3, b3 + 1, b0 + b1*c3 + c1, a0 - c3 + 2, b1*c3**2 - 2*b1*c3 + b1 - c0 + c1*c3 - 2*c1}}$ extendedgroebfactor polys; {{{b1*a0 + (b1 + c1), a2, b1*a3 + 1, a4, a5, b0 + b1, b2, b3, b4, b5, c0 + c1, c2, c3 - 1, c4, c5}, {b1,b1}, {b1,c1}}, {{a0, a2 - b0 - c1, a3, a4, a5, b0**2 + c1*b0 + c2, b1, b2, b3 + 1, b4, b5, c0, c3 - 2, c4, c5}, {}, {c1,c2}}, {{a0 + 1,a2,a3,a4,a5,b0 + (b1 + c1),b2,b3 + 1,b4,b5,c0 + c1,c2,c3 - 1,c4,c5}, {}, {b1,c1}}, {{a0 - (c3 - 2), a2, a3, a4, a5, (c3**2 - 2*c3 + 1)*b0 + (c0*c3 + c1), (c3**2 - 2*c3 + 1)*b1 - (c0 - c1*c3 + 2*c1), b2, b3 + 1, b4, b5, c2, c4, c5}, {c3**2 - 2*c3 + 1,c3**2 - 2*c3 + 1}, {c0,c1,c3}}, {{a0 - (c3 - 2), (c3 - 2)*a2 + c0 - (c1*c3 - 2*c1), a3, a4, a5, (c3 - 2)*b0 + c0, b1, b2, b3 + 1, b4, b5, c0**2 - (c1*c3 - 2*c1)*c0 + (c2*c3**2 - 4*c2*c3 + 4*c2), c4, c5}, {c3 - 2,c3 - 2}, {c1,c2,c3}}, {{(c0 - c1*c3 + 2*c1)*a0 + (c0*c3 + c1), a2, (c0 - c1*c3 + 2*c1)*a3 + (c3**2 - 2*c3 + 1), a4, a5, (c3**2 - 2*c3 + 1)*b0 - (c0*c3 - 2*c0 - c1*c3**2 + 4*c1*c3 - 4*c1), (c3**2 - 2*c3 + 1)*b1 - (c0 - c1*c3 + 2*c1), b2, b3, b4, b5, c2, c4, c5}, {c0 - c1*c3 + 2*c1, c0 - c1*c3 + 2*c1, c3**2 - 2*c3 + 1, c3**2 - 2*c3 + 1}, {c0,c1,c3}}}$ extendedgroebfactor1 polys; {{{(c0 - c1*c3 + 2*c1)*a0 + (c0*c3 + c1), a2, (c0 - c1*c3 + 2*c1)*a3 + (c3**2 - 2*c3 + 1), a4, a5, (c3**2 - 2*c3 + 1)*b0 - (c0*c3 - 2*c0 - c1*c3**2 + 4*c1*c3 - 4*c1), (c3**2 - 2*c3 + 1)*b1 - (c0 - c1*c3 + 2*c1), b2, b3, b4, b5, c2, c4, c5}, {c0 - c1*c3 + 2*c1, c0 - c1*c3 + 2*c1, c3**2 - 2*c3 + 1, c3**2 - 2*c3 + 1}, {c0,c1,c3}}, {{a0 - (c3 - 2), (c3 - 2)*a2 + c0 - (c1*c3 - 2*c1), a3, a4, a5, (c3 - 2)*b0 + c0, b1, b2, b3 + 1, b4, b5, c0**2 - (c1*c3 - 2*c1)*c0 + (c2*c3**2 - 4*c2*c3 + 4*c2), c4, c5}, {c3 - 2,c3 - 2}, {c1,c2,c3}}, {{a0 - (c3 - 2), a2, a3, a4, a5, (c3**2 - 2*c3 + 1)*b0 + (c0*c3 + c1), (c3**2 - 2*c3 + 1)*b1 - (c0 - c1*c3 + 2*c1), b2, b3 + 1, b4, b5, c2, c4, c5}, {c3**2 - 2*c3 + 1,c3**2 - 2*c3 + 1}, {c0,c1,c3}}}$ % Schwarz' example s5 vars:=for k:=1:5 collect mkid(x,k); vars := {x1, x2, x3, x4, x5}$ s5:={ x1**2+x1+2*x2*x5+2*x3*x4, 2*x1*x2+x2+2*x3*x5+x4**2, 2*x1*x3+x2**2+x3+2*x4*x5, 2*x1*x4+2*x2*x3+x4+x5**2, 2*x1*x5+2*x2*x4+x3**2+x5}; s5 := {x1**2 + x1 + 2*x2*x5 + 2*x3*x4, 2*x1*x2 + x2 + 2*x3*x5 + x4**2, 2*x1*x3 + x2**2 + x3 + 2*x4*x5, 2*x1*x4 + 2*x2*x3 + x4 + x5**2, 2*x1*x5 + 2*x2*x4 + x3**2 + x5}$ setring(vars,degreeorder vars,revlex); {{x1,x2,x3,x4,x5},{{1,1,1,1,1}},revlex,{1,1,1,1,1}}$ m:=groebfactor s5; m := {{x1**2 + 2*x3*x4 + 2*x2*x5 + x1, 2*x1*x2 + x4**2 + 2*x3*x5 + x2, x2**2 + 2*x1*x3 + 2*x4*x5 + x3, 2*x2*x3 + 2*x1*x4 + x5**2 + x4, x3**2 + 2*x2*x4 + 2*x1*x5 + x5, 2*x1*x3*x4 + 2*x4**2*x5 + x3*x5**2 + x3*x4, 5*x4**3 + 30*x3*x4*x5 + 15*x2*x5**2 - 2*x5, 10*x3*x4**2 - 10*x1*x5**2 - 5*x5**2 - x4, 625*x4*x5**3 + 50*x3*x4 + 75*x2*x5 - 6, 15*x2*x4**2 + 30*x1*x4*x5 + 5*x5**3 + 15*x4*x5 + x3, 100*x1*x4*x5**2 + 25*x5**4 + 50*x4*x5**2 + x4**2 + 4*x3*x5, 1250*x1*x3*x5**2 + 625*x3*x5**2 - 75*x3*x4 - 50*x2*x5 + 8, 75*x4**2*x5**2 + 50*x3*x5**3 + x2*x4 + 4*x1*x5 + 2*x5, 150*x3*x4*x5**2 + 100*x2*x5**3 - 2*x1*x4 - 13*x5**2 - x4, 625*x2*x5**4 - 50*x1*x4*x5 - 75*x5**3 - 25*x4*x5 - x3, 1250*x3*x5**4 - 200*x2*x4*x5 - 50*x1*x5**2 - 25*x5**2 + 3*x4, 625*x5**5 + 375*x4**2*x5 + 500*x3*x5**2 + 24*x1 + 12, 10*x1*x4**2 + 20*x1*x3*x5 + 20*x4*x5**2 + 5*x4**2 + 10*x3*x5 + x2, 75*x2*x4*x5**2 + 50*x1*x5**3 + 25*x5**3 - 2*x1*x3 - 3*x4*x5 - x3, 1250*x1*x5**4 + 625*x5**4 + 100*x1*x3*x5 + 150*x4*x5**2 + 50*x3*x5 + 3* x2}, {x5,x2,x4,x3,x1}, {x5,x2,x4,x3,x1 + 1}}$ % Recompute a list of problems with listgroebfactor for another term % order. setring(vars,{},lex); {{x1,x2,x3,x4,x5},{},lex,{1,1,1,1,1}}$ listgroebfactor m; {{5*x5 - 1, 5*x4 - 1, 5*x1 + 4, 5*x2 - 1, 5*x3 - 1}, {5*x5 + 1, 5*x4 + 1, 5*x1 + 1, 5*x2 + 1, 5*x3 + 1}, {5*x1 + 2, x2 - x5, 25*x5**2 - 5*x5 - 1, 5*x4 + 5*x5 - 1, 5*x3 + 5*x5 - 1}, {5*x1 + 3, x2 - x5, 25*x5**2 + 5*x5 - 1, 5*x4 + 5*x5 + 1, 5*x3 + 5*x5 + 1}, {5*x1 + 3, 5*x4 - 25*x5**2 + 10*x5 - 2, x3 - 25*x5**3 + 15*x5**2 - 3*x5, 5*x2 + 125*x5**3 - 50*x5**2 + 10*x5 - 1, 625*x5**4 - 375*x5**3 + 100*x5**2 - 10*x5 + 1}, {5*x1 + 2, 5*x2 + 5*x5 - 1, 5*x4 - 250*x5**3 + 75*x5**2 - 30*x5 + 2, 5*x3 + 250*x5**3 - 75*x5**2 + 30*x5 - 3, 625*x5**4 - 250*x5**3 + 100*x5**2 - 15*x5 + 1}, {x4 + 5*x5**2, 5*x1 + 1, x3 - 25*x5**3, 5*x2 + 125*x5**3 - 25*x5**2 + 5*x5 - 1, 625*x5**4 - 125*x5**3 + 25*x5**2 - 5*x5 + 1}, {x4 - 5*x5**2, 5*x1 + 4, x3 - 25*x5**3, 5*x2 + 125*x5**3 + 25*x5**2 + 5*x5 + 1, 625*x5**4 + 125*x5**3 + 25*x5**2 + 5*x5 + 1}, {5*x1 + 3, 5*x2 + 5*x5 + 1, 5*x4 - 250*x5**3 - 75*x5**2 - 30*x5 - 2, 5*x3 + 250*x5**3 + 75*x5**2 + 30*x5 + 3, 625*x5**4 + 250*x5**3 + 100*x5**2 + 15*x5 + 1}, {5*x1 + 2, 5*x4 + 25*x5**2 + 10*x5 + 2, x3 - 25*x5**3 - 15*x5**2 - 3*x5, 5*x2 + 125*x5**3 + 50*x5**2 + 10*x5 + 1, 625*x5**4 + 375*x5**3 + 100*x5**2 + 10*x5 + 1}, {x5, x2, x4, x3, x1 + 1}, {x5, x2, x4, x3, x1}}$ % ==> Testing the linear algebra package % Find the ideal of points in affine and projective space. vars:=for k:=1:6 collect mkid(x,k); vars := {x1, x2, x3, x4, x5, x6}$ setring(vars,degreeorder vars,revlex); {{x1,x2,x3,x4,x5,x6},{{1,1,1,1,1,1}},revlex,{1,1,1,1,1,1}}$ matrix mm(10,6); on rounded; for k:=1:6 do for l:=1:10 do mm(l,k):=floor(exp((k+l)/4)); off rounded; mm; mat((1,2,2,3,4,5),(2,2,3,4,5,7),(2,3,4,5,7,9),(3,4,5,7,9,12),(4,5,7,9,12,15),(5 ,7,9,12,15,20),(7,9,12,15,20,25),(9,12,15,20,25,33),(12,15,20,25,33,42),(15,20, 25,33,42,54))$ setideal(u,affine_points mm); {48337*x5**2 - 318*x4*x6 - 75336*x5*x6 + 29579*x6**2 - 11598*x1 - 11016*x2 - 11352*x3 - 2502*x4 + 18371*x5 - 1837*x6 - 1836, 386696*x1*x6 + 175678*x4*x6 + 20108*x5*x6 - 233347*x6**2 - 5821074*x1 - 831000 *x2 + 1382952*x3 + 741984*x4 - 2153934*x5 + 2692351*x6 - 1491936, 386696*x3*x6 + 239238*x4*x6 - 185716*x5*x6 - 182039*x6**2 - 270738*x1 - 1255800*x2 - 5052384*x3 - 2106864*x4 + 2351946*x5 + 2434483*x6 - 1562736, 48337*x4**2 - 58746*x4*x6 + 148*x5*x6 + 17725*x6**2 + 2502*x1 + 576*x2 - 1302* x3 + 25721*x4 - 3488*x5 - 12857*x6 + 96, 193348*x4*x5 - 151394*x4*x6 - 118604*x5*x6 + 92869*x6**2 + 1590*x1 + 8712*x2 + 16560*x3 - 3708*x4 + 43918*x5 - 43409*x6 + 1452, 386696*x2*x6 - 382090*x4*x6 - 43364*x5*x6 + 123645*x6**2 - 1313130*x1 - 4809120*x2 + 91464*x3 + 5853096*x4 + 1118658*x5 - 2323361*x6 - 28128, 193348*x6**3 - 78919578*x4*x6 - 63413004*x5*x6 + 81565689*x6**2 + 1412352942* x1 + 1563160200*x2 + 761482008*x3 + 10324224*x4 + 304232130*x5 - 1255484065*x6 - 643375200, 193348*x1*x4 - 3606*x4*x6 + 6664*x5*x6 - 36689*x6**2 - 1729374*x1 - 256248*x2 + 422132*x3 + 345556*x4 - 671778*x5 + 747089*x6 - 429404, 193348*x3*x4 - 19782*x4*x6 - 57060*x5*x6 + 1407*x6**2 - 86718*x1 - 378840*x2 - 1475924*x3 - 576996*x4 + 715078*x5 + 671885*x6 - 449836, 386696*x1*x5 + 139942*x4*x6 - 99156*x5*x6 - 94107*x6**2 - 4388370*x1 - 668088* x2 + 1063040*x3 + 468112*x4 - 1464774*x5 + 1964239*x6 - 1078088, 386696*x3*x5 + 184150*x4*x6 - 329484*x5*x6 + 3733*x6**2 - 302634*x1 - 1062840* x2 - 3845096*x3 - 1562608*x4 + 1956858*x5 + 1752663*x6 - 1143880, 193348*x1**2 + 50726*x4*x6 + 4164*x5*x6 - 49995*x6**2 - 1502518*x1 - 234624*x2 + 337000*x3 + 189344*x4 - 544926*x5 + 709519*x6 - 425800, 193348*x1*x2 - 22834*x4*x6 - 3056*x5*x6 - 4123*x6**2 - 1183010*x1 - 780060*x2 + 282940*x3 + 989552*x4 - 238902*x5 + 102179*x6 - 226684, 386696*x1*x3 + 153254*x4*x6 - 46332*x5*x6 - 109011*x6**2 - 2706290*x1 - 776040 *x2 - 650592*x3 - 288096*x4 - 295470*x5 + 1856303*x6 - 1096080, 96674*x3**2 + 56278*x4*x6 - 44916*x5*x6 - 20423*x6**2 - 85218*x1 - 315900*x2 - 1104614*x3 - 478348*x4 + 559514*x5 + 528787*x6 - 342672, 193348*x2*x4 - 186922*x4*x6 - 13484*x5*x6 + 80823*x6**2 - 400398*x1 - 1437268* x2 + 32400*x3 + 1825348*x4 + 346526*x5 - 749039*x6 - 13972, 386696*x2*x5 - 299774*x4*x6 - 183476*x5*x6 + 214259*x6**2 - 1032390*x1 - 3818088*x2 + 38568*x3 + 4563624*x4 + 1175646*x5 - 2005927*x6 - 56304, 193348*x5*x6**2 - 59111766*x4*x6 - 58092824*x5*x6 + 68856463*x6**2 + 1134803514*x1 + 1224865560*x2 + 591403776*x3 - 36549312*x4 + 316517430*x5 - 1023460331*x6 - 498675720, 96674*x2**2 - 69590*x4*x6 - 7908*x5*x6 + 35327*x6**2 - 243426*x1 - 832910*x2 + 14700*x3 + 1041208*x4 + 204662*x5 - 420851*x6 - 26032, 386696*x2*x3 - 95202*x4*x6 - 92692*x5*x6 + 63421*x6**2 - 736122*x1 - 2670472* x2 - 1669344*x3 + 2061800*x4 + 1466002*x5 - 394913*x6 - 509528, 96674*x4*x6**2 - 29277416*x4*x6 - 19752504*x5*x6 + 28388673*x6**2 + 433544670* x1 + 473916360*x2 + 235723620*x3 + 39729120*x4 + 97903830*x5 - 411516489*x6 - 188800920}$ setgbasis u$ dim u; 0$ degree u; 10$ setideal(u,proj_points mm); {457380*x5**3 - 13500*x2*x5*x6 - 20835*x3*x5*x6 + 76950*x4*x5*x6 - 1050234*x5** 2*x6 + 100*x1*x6**2 + 10800*x2*x6**2 + 16568*x3*x6**2 - 60271*x4*x6**2 + 771366*x5*x6**2 - 179875*x6**3, 330*x4**2 + 1665*x2*x5 + 555*x3*x5 - 4337*x4*x5 - 626*x5**2 + 6*x1*x6 - 1332* x2*x6 - 450*x3*x6 + 3013*x4*x6 + 2740*x5*x6 - 1635*x6**2, 33*x1*x5 - 90*x2*x5 - 63*x3*x5 + 216*x4*x5 + 60*x5**2 - 25*x1*x6 + 72*x2*x6 + 49*x3*x6 - 170*x4*x6 - 171*x5*x6 + 97*x6**2, 90*x3**2 - 483*x2*x5 - 183*x3*x5 + 1197*x4*x5 + 174*x5**2 - 2*x1*x6 + 384*x2* x6 + 68*x3*x6 - 937*x4*x6 - 738*x5*x6 + 485*x6**2, 330*x3*x4 + 555*x2*x5 + 75*x3*x5 - 1519*x4*x5 - 172*x5**2 + 2*x1*x6 - 444*x2* x6 - 260*x3*x6 + 1041*x4*x6 + 950*x5*x6 - 545*x6**2, 457380*x4*x5**2 - 10800*x2*x5*x6 - 9045*x3*x5*x6 - 662625*x4*x5*x6 - 265413*x5 **2*x6 + 80*x1*x6**2 + 8640*x2*x6**2 + 7156*x3*x6**2 + 238408*x4*x6**2 + 391452*x5*x6**2 - 143900*x6**3, 495*x1*x2 + 1977*x2*x5 + 87*x3*x5 - 4824*x4*x5 - 339*x5**2 - 164*x1*x6 - 1707* x2*x6 - 97*x3*x6 + 3782*x4*x6 + 2697*x5*x6 - 1840*x6**2, 495*x2**2 + 1134*x2*x5 + 939*x3*x5 - 3771*x4*x5 - 822*x5**2 - 4*x1*x6 - 1257* x2*x6 - 734*x3*x6 + 2956*x4*x6 + 2709*x5*x6 - 1550*x6**2, 990*x1*x3 - 5043*x2*x5 - 1263*x3*x5 + 12321*x4*x5 + 1866*x5**2 - 464*x1*x6 + 4008*x2*x6 + 788*x3*x6 - 9643*x4*x6 - 7968*x5*x6 + 5165*x6**2, 495*x2*x3 + 1242*x2*x5 - 48*x3*x5 - 3555*x4*x5 - 267*x5**2 + 4*x1*x6 - 1218*x2 *x6 - 157*x3*x6 + 2786*x4*x6 + 2142*x5*x6 - 1420*x6**2, 66*x1*x4 + 345*x2*x5 + 93*x3*x5 - 861*x4*x5 - 120*x5**2 - 38*x1*x6 - 276*x2*x6 - 76*x3*x6 + 659*x4*x6 + 540*x5*x6 - 337*x6**2, 495*x2*x4 + 1770*x2*x5 + 645*x3*x5 - 4908*x4*x5 - 729*x5**2 + 4*x1*x6 - 1713* x2*x6 - 520*x3*x6 + 3677*x4*x6 + 3165*x5*x6 - 1915*x6**2, 152460*x3*x5**2 + 5880*x2*x5*x6 - 237983*x3*x5*x6 - 6896*x4*x5*x6 - 71438*x5** 2*x6 + 64*x1*x6**2 - 4704*x2*x6**2 + 92748*x3*x6**2 + 5427*x4*x6**2 + 113560* x5*x6**2 - 45061*x6**3, 990*x1**2 - 1893*x2*x5 + 447*x3*x5 + 3771*x4*x5 + 426*x5**2 - 524*x1*x6 + 1488 *x2*x6 - 322*x3*x6 - 2923*x4*x6 - 2478*x5*x6 + 1715*x6**2, 457380*x2*x5**2 - 754524*x2*x5*x6 - 31530*x3*x5*x6 + 156561*x4*x5*x6 - 132597* x5**2*x6 + 136*x1*x6**2 + 310896*x2*x6**2 + 25088*x3*x6**2 - 122581*x4*x6**2 + 141732*x5*x6**2 - 30097*x6**3}$ setgbasis u$ dim u; 1$ degree u; 10$ % Change the term order to pure lex in dimension zero. % Test both approaches, with and without precomputed borderbasis. vars:=for k:=1:6 collect mkid(x,k); vars := {x1, x2, x3, x4, x5, x6}$ r1:=setring(vars,{},lex); r1 := {{x1,x2,x3,x4,x5,x6},{},lex,{1,1,1,1,1,1}}$ r2:=setring(vars,degreeorder vars,revlex); r2 := {{x1,x2,x3,x4,x5,x6},{{1,1,1,1,1,1}},revlex,{1,1,1,1,1,1}}$ setideal(m,{x1**2+x1+2*x2*x6+2*x3*x5+x4**2, 2*x1*x2+x2+2*x3*x6+2*x4*x5, 2*x1*x3+x2**2+x3+2*x4*x6+x5**2, 2*x1*x4+2*x2*x3+x4+2*x5*x6, 2*x1*x5+2*x2*x4+x3**2+x5+x6**2, 2*x1*x6+2*x2*x5+2*x3*x4+x6}); {x1**2 + x4**2 + 2*x3*x5 + 2*x2*x6 + x1, 2*x1*x2 + 2*x4*x5 + 2*x3*x6 + x2, x2**2 + 2*x1*x3 + x5**2 + 2*x4*x6 + x3, 2*x2*x3 + 2*x1*x4 + 2*x5*x6 + x4, x3**2 + 2*x2*x4 + 2*x1*x5 + x6**2 + x5, 2*x3*x4 + 2*x2*x5 + 2*x1*x6 + x6}$ gbasis m; {72*x1*x3*x5*x6 + 36*x3*x5*x6 - 2*x1*x6 - x6, 2*x1*x2 + 2*x4*x5 + 2*x3*x6 + x2, 2*x2*x3 + 2*x1*x4 + 2*x5*x6 + x4, 2*x3*x4 + 2*x2*x5 + 2*x1*x6 + x6, 10368*x6**7 + 5040*x4*x6**4 + 140*x4**2*x6 + 252*x2*x6**2 - 15*x6, 1296*x4*x6**5 + 180*x4**2*x6**2 + 180*x2*x6**3 + 4*x2*x4 - 15*x6**2, 2592*x3*x6**5 - 360*x2*x5*x6**2 + 2*x1*x4 + 12*x5*x6 + x4, 72*x3*x5**2*x6 + 72*x2*x5*x6**2 - 2*x1*x4 - 8*x5*x6 - x4, 36*x2*x5**3 - 108*x1*x4*x6**2 - 72*x5*x6**3 - 54*x4*x6**2 - 5*x3*x6, 18*x4**2*x5 + 18*x3*x5**2 - 18*x1*x6**2 - 9*x6**2 - 2*x5, 36*x4*x5**2 + 36*x4**2*x6 + 72*x3*x5*x6 + 36*x2*x6**2 - 5*x6, x1**2 + x4**2 + 2*x3*x5 + 2*x2*x6 + x1, x2**2 + 2*x1*x3 + x5**2 + 2*x4*x6 + x3, x3**2 + 2*x2*x4 + 2*x1*x5 + x6**2 + x5, 2592*x5*x6**5 + 360*x4*x5*x6**2 + 360*x3*x6**3 - 2*x2*x5 + 10*x1*x6 + 5*x6, 2592*x5**2*x6**3 + 1296*x4*x6**4 + 36*x4**2*x6 + 144*x3*x5*x6 + 180*x2*x6**2 - 13*x6, 72*x5**3*x6 + 216*x4*x5*x6**2 + 72*x3*x6**3 + 2*x2*x5 + 6*x1*x6 + 3*x6, 4*x4**3 - 12*x2*x5**2 - 24*x1*x5*x6 - 4*x6**3 - 12*x5*x6 - x4, 12*x2*x4**2 - 24*x1*x3*x6 - 12*x5**2*x6 - 12*x4*x6**2 - 12*x3*x6 - x2, 1296*x1*x6**5 + 648*x6**5 + 180*x1*x4*x6**2 + 180*x5*x6**3 + 90*x4*x6**2 + x4* x5 + 6*x3*x6, 2592*x2*x6**5 + 360*x2*x4*x6**2 - 180*x6**4 - 6*x1*x3 - 3*x5**2 - 16*x4*x6 - 3 *x3, 72*x3*x5*x6**3 + 36*x2*x6**4 - x2*x5**2 - 4*x2*x4*x6 - 4*x1*x5*x6 - 6*x6**3 - 2*x5*x6, 648*x4*x5*x6**3 + 324*x3*x6**4 - 9*x3*x5**2 - 18*x2*x5*x6 + 18*x1*x6**2 + 9*x6 **2 + x5, 2592*x1*x3*x6**3 + 1296*x4*x6**4 + 1296*x3*x6**3 - 36*x4**2*x6 - 72*x3*x5*x6 + 36*x2*x6**2 + 5*x6, 1080*x2*x4*x6**3 + 216*x6**5 - 60*x1*x3*x6 - 30*x5**2*x6 - 90*x4*x6**2 - 30*x3 *x6 - x2, 72*x4**2*x6**3 + 36*x2*x6**4 + 3*x2*x5**2 + 8*x2*x4*x6 + 6*x1*x5*x6 - 2*x6**3 + 3*x5*x6, 36*x1*x5**2*x6 + 36*x1*x4*x6**2 + 36*x5*x6**3 + 18*x5**2*x6 + 18*x4*x6**2 + x4 *x5 + 2*x3*x6, 18*x1*x5**3 - 54*x1*x3*x6**2 - 36*x4*x6**3 + 9*x5**3 - 27*x3*x6**2 + x3*x5 - 2 *x2*x6, 18*x1*x4*x5 + 18*x1*x3*x6 + 18*x5**2*x6 + 18*x4*x6**2 + 9*x4*x5 + 9*x3*x6 + x2 , 18*x2*x4*x5 + 18*x1*x5**2 + 18*x1*x4*x6 + 18*x5*x6**2 + 9*x5**2 + 9*x4*x6 + x3 , 2*x1*x4**2 + 4*x1*x3*x5 + 2*x5**3 + 8*x4*x5*x6 + 2*x3*x6**2 + x4**2 + 2*x3*x5, 72*x1*x4*x6**3 + 36*x5*x6**4 + 36*x4*x6**3 - 2*x1*x3*x5 - x5**3 - 2*x4*x5*x6 + 2*x3*x6**2 - x3*x5, 3240*x1*x5*x6**3 + 648*x6**5 + 1620*x5*x6**3 + 90*x1*x3*x6 + 90*x5**2*x6 + 180 *x4*x6**2 + 45*x3*x6 + 2*x2, 72*x2*x5**2*x6 + 72*x2*x4*x6**2 + 144*x1*x5*x6**2 + 36*x6**4 + 72*x5*x6**2 - 2 *x1*x3 - x5**2 - x3, 36*x5**4 - 216*x4**2*x6**2 - 432*x3*x5*x6**2 - 288*x2*x6**3 + 2*x2*x4 + 8*x1* x5 + 39*x6**2 + 4*x5, 72*x3*x5**3 - 216*x1*x5*x6**2 - 36*x6**4 - 108*x5*x6**2 - 2*x1*x3 - 9*x5**2 - 8*x4*x6 - x3, 1296*x2*x5*x6**3 + 648*x1*x6**4 + 324*x6**4 - 18*x1*x5**2 - 36*x1*x4*x6 - 72* x5*x6**2 - 9*x5**2 - 18*x4*x6 - x3, 18*x1*x3*x5**2 + 18*x4**2*x6**2 + 54*x3*x5*x6**2 + 36*x2*x6**3 + 9*x3*x5**2 - x2*x4 - 2*x1*x5 - 5*x6**2 - x5}$ m1:=change_termorder(m,r1); m1 := {46656*x4*x5*x6**6 - x4*x5, 80621568*x5*x6**13 + 44928*x5*x6**7 - x5*x6, 637*x2 + 1077507256320*x6**17 - 927987840*x6**11 - 806157*x6**5, 7*x4**2*x5 + 252*x4*x5*x6**3 - 30233088*x5*x6**12 - 17496*x5*x6**6, 58773123072*x6**19 - 47869056*x6**13 - 45657*x6**7 + x6, 2*x1*x4 + x4 + 1296*x5**3*x6**3 + 3*x5*x6, 2548*x4**2*x6 + 91728*x4*x6**4 + 362797056*x6**13 - 33696*x6**7 - 141*x6 , 8491392*x4*x6**7 - 182*x4*x6 + 19591041024*x6**16 + 10917504*x6**10 - 243*x6**4, 17199*x5**4*x6 + 1238328*x5**2*x6**5 + 4353564672*x6**15 - 5003856*x6**9 - 1328*x6**3, 4245696*x5**2*x6**7 - 91*x5**2*x6 + 8707129344*x6**17 - 12130560*x6**11 + 256*x6**5, 7*x3 - 567*x5**5 - 204120*x5**3*x6**4 + 524880*x5*x6**8 - 90*x5*x6**2, 2916*x5**7 - 567*x5**3*x6**2 + 347680512*x5*x6**12 + 196668*x5*x6**6 - 4 *x5, 14*x1*x6 + 504*x4*x5*x6**2 + 252*x5**3*x6 + 20155392*x5*x6**11 + 8640*x5 *x6**5 + 7*x6, 2548*x4**3 + 19813248*x4*x6**6 - 637*x4 + 39182082048*x6**15 + 30326400* x6**9 + 4428*x6**3, 1911*x4*x5**2 - 68796*x4*x6**4 - 68796*x5**2*x6**3 - 362797056*x6**13 + 151632*x6**7 + 50*x6, 1274*x1*x5 + 1651104*x4*x6**5 + 5733*x5**4 + 1238328*x5**2*x6**4 + 637* x5 + 6530347008*x6**14 - 1667952*x6**8 + 192*x6**2, 1274*x1**2 + 1274*x1 + 1274*x4**2 + 206388*x5**6 - 85995*x5**2*x6**2 - 11754624614400*x6**18 + 9498228480*x6**12 + 9295668*x6**6}$ setring r2$ m2:=change_termorder1(m,r1); m2 := {46656*x4*x5*x6**6 - x4*x5, 80621568*x5*x6**13 + 44928*x5*x6**7 - x5*x6, 2*x1*x4 + x4 + 1296*x5**3*x6**3 + 3*x5*x6, 637*x2 + 1077507256320*x6**17 - 927987840*x6**11 - 806157*x6**5, 7*x4**2*x5 + 252*x4*x5*x6**3 - 30233088*x5*x6**12 - 17496*x5*x6**6, 58773123072*x6**19 - 47869056*x6**13 - 45657*x6**7 + x6, 2548*x4**2*x6 + 91728*x4*x6**4 + 362797056*x6**13 - 33696*x6**7 - 141*x6 , 17199*x5**4*x6 + 1238328*x5**2*x6**5 + 4353564672*x6**15 - 5003856*x6**9 - 1328*x6**3, 4245696*x5**2*x6**7 - 91*x5**2*x6 + 8707129344*x6**17 - 12130560*x6**11 + 256*x6**5, 7*x3 - 567*x5**5 - 204120*x5**3*x6**4 + 524880*x5*x6**8 - 90*x5*x6**2, 8491392*x4*x6**7 - 182*x4*x6 + 19591041024*x6**16 + 10917504*x6**10 - 243*x6**4, 2916*x5**7 - 567*x5**3*x6**2 + 347680512*x5*x6**12 + 196668*x5*x6**6 - 4 *x5, 14*x1*x6 + 504*x4*x5*x6**2 + 252*x5**3*x6 + 20155392*x5*x6**11 + 8640*x5 *x6**5 + 7*x6, 1911*x4*x5**2 - 68796*x4*x6**4 - 68796*x5**2*x6**3 - 362797056*x6**13 + 151632*x6**7 + 50*x6, 2548*x4**3 + 19813248*x4*x6**6 - 637*x4 + 39182082048*x6**15 + 30326400* x6**9 + 4428*x6**3, 1274*x1*x5 + 1651104*x4*x6**5 + 5733*x5**4 + 1238328*x5**2*x6**4 + 637* x5 + 6530347008*x6**14 - 1667952*x6**8 + 192*x6**2, 1274*x1**2 + 1274*x1 + 1274*x4**2 + 206388*x5**6 - 85995*x5**2*x6**2 - 11754624614400*x6**18 + 9498228480*x6**12 + 9295668*x6**6}$ setideal(m1,m1)$ setideal(m2,m2)$ setgbasis m1$ setgbasis m2$ modequalp(m1,m2); yes$ % ==> Different hilbert series driver setideal(m,proj_monomial_curve(w1:={0,2,5,9},{w,x,y,z})); {x**5 - w**3*y**2, w*y**3 - x**3*z, y**4 - w*x*z**2, x**2*y - w**2*z}$ weights:={{1,1,1,1},w1}; weights := {{1,1,1,1},{0,2,5,9}}$ hftestversion 2; hf!=whilb2$ f1:=weightedhilbertseries(gbasis m,weights); f1 := ( - w**5*x**17 + w**4*x**17 - w**4*x**15 + w**4*x**8 + w**3*x**15 + w**3* x**12 + w**3*x**6 + w**2*x**10 + w**2*x**7 + w**2*x**4 + w*x**5 + w*x**2 + 1)/(w**2*x**9 - w*x**9 - w + 1)$ sub(x=1,ws); ( - w**5 + w**4 + 3*w**3 + 3*w**2 + 2*w + 1)/(w**2 - 2*w + 1)$ % The ordinary Hilbert series. hftestversion 1; hf!=whilb1$ % The default. f2:=weightedhilbertseries(gbasis m,weights); f2 := ( - w**5*x**17 + w**4*x**17 - w**4*x**15 + w**4*x**8 + w**3*x**15 + w**3* x**12 + w**3*x**6 + w**2*x**10 + w**2*x**7 + w**2*x**4 + w*x**5 + w*x**2 + 1)/(w**2*x**9 - w*x**9 - w + 1)$ sub(x=1,ws); ( - w**5 + w**4 + 3*w**3 + 3*w**2 + 2*w + 1)/(w**2 - 2*w + 1)$ f1-f2; 0$ % ==> Different primary decomposition approaches. The example is due % to Shimoyama Takeshi. CALI 2.2. produced auxiliary embedded % primes on it. vars:={dx,dy,x,y}; vars := {dx,dy,x,y}$ setring(vars,degreeorder vars,revlex); {{dx,dy,x,y},{{1,1,1,1}},revlex,{1,1,1,1}}$ f3:={DY*( - X*DX + Y**2*DY - Y*DY),DX*(X**2*DX - X*DX - Y*DY)}$ primarydecomposition f3; {{{dx**3, dy**3, dx**2*dy, dx*dy**2, dy*( - dx*x + dy*y**2 - dy*y), dx*(dx*x**2 - dx*x - dy*y)}, {dx,dy}}, {{x*y - x - y, dx*x - dx - dy*y + dy, - dx + dy*y**2 - 2*dy*y + dy}, {x*y - x - y, dx*x - dx - dy*y + dy, - dx + dy*y**2 - 2*dy*y + dy}}, {{dy,x - 1},{dy,x - 1}}, {{dy**2, dy*x, x**2, dx*x + dy*y}, {dy,x}}, {{dx,y - 1},{dx,y - 1}}, {{dx**2, dx*y, y**2, dx*x + dy*y}, {dx,y}}, {{y**2, x**2, x*y, dx*x + dy*y}, {y,x}}}$ showtime; Time: 277500 ms plus GC time: 10350 ms end; (TIME: cali 277530 287880)