% ;; Problem: find a minimal standard basis of an inhomogeneous ideal. % ;; % ;; The technique we use is to homogenize the ideal, compute a standard % ;; basis, and then find a minimal standard basis after dehomogenizing. % ring R ! characteristic (if not 31991) ? <return> ! number of variables ? 5 ! 5 variables, please ? a-dt ! variable weights (if not all 1) ? <return> ! monomial order (if not rev. lex.) ? 4 1 ; largest degree of a monomial : 512 512 % ideal j ! number of generators ? 3 ! (1,1) ? a3-bc-1 ! (1,2) ? ab2-cd-a ! (1,3) ? b3-bc-d+1 % ; now we homogenize % homog j t k % type k ; a3-bct-t3 ab2-cdt-at2 b3-bct-dt2+t3 % ; compute a standard basis of "k" % std k k ; 34.5.6.7.8.9. ; computation complete after degree 9 % ; find a subset of this standard basis which is minimal after % ; dehomogenization % stdpart k m a..d % putmat m ; 1 ; 11 ; c3d3t2-bc4t3+b2c2t4+3bc3t4-3c3dt4-3bcd2t4-2ac2t5-3bc2t5+c3t5 \ ; +abdt5+2bcdt5+8c2dt5+2ad2t5-abt6-3b2t6+5act6+2bct6-3c2t6 \ ; -4adt6+3bdt6-6cdt6-d2t6-at7-3bt7+3ct7+2dt7 ; bcd3t-b2c2t2-ac2dt2-a2d2t2+abd2t2+c2d2t2-ad3t2+a2bt3-a2ct3 \ ; +b2ct3+a2dt3+2acdt3-bcdt3+ad2t3+a2t4+bt5-dt5+t6 ; b2cdt-abdt2-c2dt2+abt3-act3+cdt3+at4 ; b3-bct-dt2+t3 ; ac2d2t2-bc3t3+2bc2t4-abdt4-2c2dt4+abt5+2b2t5-act5-bct5+c2t5 \ ; -bdt5+3cdt5+at6+bt6-2ct6-t7 ; ac3dt2-c3d2t2+a2c2t3+b2c2t3-2ac2dt3-bcd2t3-a2bt4-a2ct4-b2ct4 \ ; +a2dt4-abdt4+bcdt4+ad2t4-a2t5-adt5-bt6+dt6-t7 ; abct-bcdt-abt2+adt2-at3 ; ab2-cdt-at2 ; a2cdt-bc2t2-b2t3+bct3-cdt3+ct4+t5 ; a2bdt2+ac2dt2-c2d2t2-a2bt3+a2ct3-2acdt3-a2t4 ; a3-bct-t3 % ; finally, dehomogenize % imap f m m ! variable, new image ? t 1 ! variable, new image ? <return> % ev f m m % ; and the final answer is % putmat m ; 1 ; 11 ; c3d3-bc4+b2c2+3bc3-3c3d-3bcd2-2ac2-3bc2+c3+abd+2bcd+8c2d+2ad2 \ ; -ab-3b2+5ac+2bc-3c2-4ad+3bd-6cd-d2-a-3b+3c+2d ; bcd3-b2c2-ac2d-a2d2+abd2+c2d2-ad3+a2b-a2c+b2c+a2d+2acd-bcd+ad2 \ ; +a2+b-d+1 ; b2cd-abd-c2d+ab-ac+cd+a ; b3-bc-d+1 ; ac2d2-bc3+2bc2-abd-2c2d+ab+2b2-ac-bc+c2-bd+3cd+a+b-2c-1 ; ac3d-c3d2+a2c2+b2c2-2ac2d-bcd2-a2b-a2c-b2c+a2d-abd+bcd+ad2-a2 \ ; -ad-b+d-1 ; abc-bcd-ab+ad-a ; ab2-cd-a ; a2cd-bc2-b2+bc-cd+c+1 ; a2bd+ac2d-c2d2-a2b+a2c-2acd-a2 ; a3-bc-1