Ask Your Question

Revision history [back]

click to hide/show revision 1
initial version

The best option is to just use I.groebner_basis().

sage: P.<a,b,c,e,f,g,h,i,j,k> = PolynomialRing(GF(32003),10)
sage: I = sage.rings.ideal.Katsura(P,6)
sage: I.groebner_basis()
Polynomial Sequence with 22 Polynomials in 6 Variables
sage: list(_)
[g^6 - 11466*g^5 - 3669*b*g^3 + 4465*c*g^3 - 2315*e*g^3 + 5098*f*g^3 - 8372*g^4 - 15837*f^3 - 9547*b*e*g + 839*b*f*g + 15210*c*f*g - 2141*f^2*g - 13725*b*g^2 - 10720*c*g^2 - 8122*e*g^2 - 2102*f*g^2 - 13908*g^3 + 4663*b*e + 12265*b*f + 6285*c*f + 12921*e*f + 11846*f^2 + 6867*b*g + 9878*c*g - 860*e*g - 1743*f*g - 8368*g^2 + 1892*b + 8121*c - 3112*e + 4349*f + 2284*g, b*g^4 + 12019*g^5 + 6465*b*g^3 - 13833*c*g^3 - 14150*e*g^3 - 1091*f*g^3 - 14524*g^4 - 15914*f^3 - 15812*b*e*g + 15278*b*f*g - 4542*c*f*g - 7300*f^2*g + 9977*b*g^2 - 14014*c*g^2 + 3296*e*g^2 + 5812*f*g^2 - 10065*g^3 - 4292*b*e - 463*b*f - 12423*c*f - 853*e*f - 11889*f^2 + 3839*b*g + 6212*c*g - 7338*e*g + 1142*f*g + 12073*g^2 + 15253*b - 7437*c + 11862*e - 6614*f + 11312*g, c*g^4 - 13467*g^5 + 14690*b*g^3 - 1071*c*g^3 + 7136*e*g^3 + 249*f*g^3 + 5472*g^4 + 11550*f^3 + 3557*b*e*g + 1925*b*f*g + 9868*c*f*g - 130*f^2*g - 3432*b*g^2 - 6768*c*g^2 + 1982*e*g^2 + 12826*f*g^2 - 12180*g^3 + 13808*b*e + 11322*b*f + 2171*c*f + 12372*e*f + 10928*f^2 + 750*b*g + 9052*c*g - 9008*e*g - 2003*f*g - 5879*g^2 - 15186*b + 8220*c - 6034*e - 7337*f + 906*g, e*g^4 - 13178*g^5 - 2372*b*g^3 + 2247*c*g^3 - 14789*e*g^3 + 2351*f*g^3 - 6361*g^4 + 8004*f^3 - 3036*b*e*g + 11459*b*f*g - 15592*c*f*g - 1543*f^2*g - 903*b*g^2 - 9248*c*g^2 - 10163*e*g^2 - 10066*f*g^2 - 6929*g^3 + 8587*b*e + 3419*b*f - 12100*c*f - 6787*e*f - 10573*f^2 - 2036*b*g + 8698*c*g + 7884*e*g - 7506*f*g + 2111*g^2 + 14794*b - 15398*c + 13481*e - 3213*f + 2440*g, f*g^4 - 3764*g^5 - 14119*b*g^3 + 2796*c*g^3 + 11676*e*g^3 - 3057*f*g^3 + 12155*g^4 + 3551*f^3 - 15434*b*e*g - 1738*b*f*g - 8112*c*f*g - 15180*f^2*g + 8238*b*g^2 - 10096*c*g^2 + 311*e*g^2 - 3845*f*g^2 + 8868*g^3 - 15590*b*e - 10226*b*f - 7332*c*f - 1428*e*f - 674*f^2 + 2801*b*g + 3547*c*g - 3307*e*g + 5875*f*g - 12862*g^2 - 12845*b - 5085*c - 12889*e - 15681*f - 2238*g, f^4 - 6399*b*g^3 - 2550*c*g^3 - 4250*e*g^3 - 8085*f*g^3 - 7747*g^4 - 1422*f^3 + 8961*b*e*g + 2851*b*f*g - 13025*c*f*g - 9339*f^2*g + 7608*b*g^2 - 13449*c*g^2 - 1005*e*g^2 - 9444*f*g^2 + 11939*g^3 + 2327*b*e + 13120*b*f + 1164*c*f - 9780*e*f + 491*f^2 - 12148*b*g + 8222*c*g - 2408*e*g - 12213*f*g + 2945*g^2 + 7016*b + 11178*c + 8965*e + 15869*f - 836*g, f^3*g - 2*c*g^3 + 2125*e*g^3 + 14315*f*g^3 + 600*g^4 - 997*f^3 + 4267*b*f*g - 1575*c*f*g + 15077*f^2*g - 10190*b*g^2 - 7880*c*g^2 - 3973*e*g^2 + 11365*f*g^2 - 4299*g^3 + 15971*b*e - 9035*b*f + 4557*c*f - 10984*e*f + 3217*f^2 + 7090*b*g + 10122*c*g + 614*e*g + 9456*f*g + 14192*g^2 + 2981*b - 12038*c - 1883*e - 9555*f - 11387*g, b*e*g^2 + 5975*b*g^3 - 12203*c*g^3 - 1563*e*g^3 + 14282*f*g^3 + 1566*g^4 + 4353*f^3 - 1451*b*e*g + 14986*b*f*g - 402*c*f*g + 9649*f^2*g + 487*b*g^2 + 8580*c*g^2 - 6371*e*g^2 - 15445*f*g^2 + 7780*g^3 - 3975*b*e - 14275*b*f + 11511*c*f + 13013*e*f - 8885*f^2 - 5289*b*g - 14943*c*g + 2915*e*g + 3362*f*g + 15818*g^2 + 7326*b + 10249*c + 12470*e - 13007*f - 9751*g, b*f*g^2 - 9388*b*g^3 + 8448*c*g^3 + 8959*e*g^3 - 7427*f*g^3 + 14506*g^4 - 8706*f^3 - 13356*b*e*g + 7648*b*f*g - 15734*c*f*g + 12354*f^2*g - 14107*b*g^2 + 5462*c*g^2 + 3279*e*g^2 - 222*f*g^2 + 4397*g^3 - 9530*b*e + 10918*b*f - 13817*c*f + 15086*e*f - 5708*f^2 + 7733*b*g - 500*c*g - 2831*e*g + 5751*f*g + 11927*g^2 + 7521*b + 10711*c + 1481*e + 15651*f + 925*g, c*f*g^2 + 12802*b*g^3 + 15789*c*g^3 + 4979*e*g^3 + 3130*f*g^3 + 5690*g^4 - 15397*f^3 - 7254*b*e*g - 2002*b*f*g + 15449*c*f*g + 7902*f^2*g + 4037*b*g^2 - 3325*c*g^2 + 4380*e*g^2 + 7239*f*g^2 + 10261*g^3 + 3579*b*e - 12205*b*f - 6288*c*f + 11795*e*f + 12751*f^2 - 15818*b*g + 4520*c*g + 14852*e*g - 9197*f*g - 2169*g^2 - 7202*b + 10510*c - 8112*e + 4780*f + 14781*g, f^2*g^2 + 2*e*g^3 + 14939*f*g^3 + 6404*g^4 + 5965*f^3 + 8303*f^2*g + 4267*b*g^2 + 5138*c*g^2 + 15432*e*g^2 + 11769*f*g^2 - 3273*g^3 + 3787*b*f + 1609*c*f - 4954*e*f + 13731*f^2 - 9712*b*g + 1829*c*g - 7181*e*g - 10184*f*g + 11056*g^2 - 1975*b + 7110*c + 9480*e + 11060*f + 11850*g, b*e*f + 1641*f^3 + 2*b*e*g - 6154*b*f*g - 4923*c*f*g - 4036*f^2*g + 7388*b*g^2 + 1232*c*g^2 - 8069*e*g^2 - 13606*f*g^2 - 918*g^3 - 10873*b*e - 1231*b*f - 15181*c*f + 14358*e*f - 5132*f^2 + 5110*b*g - 6640*c*g - 14531*e*g - 12203*f*g - 11339*g^2 + 7443*b - 10062*c - 12527*e - 12324*f - 6786*g, b*f^2 - 3282*f^3 + 2*b*e*g - 2464*b*f*g + 7382*c*f*g - 12760*f^2*g - 4922*b*g^2 - 9845*c*g^2 + 6489*e*g^2 + 15478*f*g^2 + 14720*g^3 - 5334*b*e - 11078*b*f - 1641*c*f + 8821*e*f - 1129*f^2 - 12115*b*g - 14610*c*g + 13849*e*g + 13771*f*g - 1920*g^2 + 6950*b - 14055*c - 3901*e - 3081*f + 13228*g, c*f^2 + 7385*f^3 - 16000*b*f*g + 7388*c*f*g - 4998*f^2*g + 12310*b*g^2 + 3694*c*g^2 + 7238*e*g^2 + 505*f*g^2 - 2713*g^3 + 5539*b*e - 11078*b*f + 8616*c*f - 4925*e*f + 9845*f^2 + 11493*b*g - 9419*c*g + 1886*e*g + 2942*f*g - 705*g^2 + 10743*b - 4930*c - 11702*e + 11068*f + 14760*g, e*f^2 - 9846*f^3 + c*f*g - 9117*f^2*g - 12311*c*g^2 + 3915*e*g^2 - 2639*f*g^2 - 7374*g^3 - 11078*b*f - 12309*c*f + 6392*e*f + 13896*f^2 - 9075*b*g + 5788*c*g + 12586*e*g - 9607*f*g + 15568*g^2 - 15170*b - 2622*c - 3496*e + 6589*f - 4370*g, e*f*g - 9846*f^2*g + c*g^2 + 12311*e*g^2 + 4927*f*g^2 + 4926*g^3 - 2790*e*f - 4185*f^2 - 11078*b*g - 15099*c*g - 12063*e*g - 1970*f*g - 2873*g^2 - 6975*b - 12555*c + 15263*e + 12473*f + 11078*g, b^2 + 2*b*e - 2*c*f - 5822*e*f - 8731*f^2 - 5822*c*g + 14543*e*g - 2917*f*g + 2905*g^2 - 14547*b + 5819*c - 2909*e - 8727*f - 11636*g, b*c - 2*b*e + 3*b*f + 4*c*f - 2905*e*f + 11642*f^2 - 2906*c*g - 8725*e*g + 14555*f*g - 14543*g^2 + 8728*b + 2909*c - 1455*e + 11636*f - 5820*g, c^2 + 2*b*e - 4*b*f - 4*c*f - 4*e*f - 4*f^2 + 2*b*g - 4*f*g + f, c*e + b*f - 14548*e*f - 5820*f^2 - 2*b*g - 14550*c*g - 11642*e*g + 8722*f*g - 8733*g^2 - 4364*b + 14547*c - 7273*e - 5818*f + 2911*g, e^2 + 2*c*f - 2905*e*f + 11641*f^2 + 2*b*g - 2905*c*g - 8721*e*g + 14557*f*g - 14540*g^2 + 8728*b + 2909*c + 14546*e + 11636*f - 5821*g, a + 2*b + 2*c + 2*e + 2*f + 2*g - 1]

Did you get an error when running that? What version of Sage were you using?