2003.sagews

R = QQ[t_1..t_3,x_1..x_4,MonomialOrder=>Lex]
use R
I = ideal (x_1-t_1^3*t_2^4*t_3^2,x_2-t_1^2*t_2^3*t_3^4,x_3-t_1^1*t_2^1*t_3^2,x_4-t_1^2*t_3)
Igb= gb I
netList entries transpose gens Igb

R PolynomialRing R PolynomialRing 3 4 2 2 3 4 2 2 ideal (- t t t + x , - t t t + x , - t t t + x , - t t + x ) 1 2 3 1 1 2 3 2 1 2 3 3 1 3 4 Ideal of R GroebnerBasis[status: done; S-pairs encountered up to degree 34] GroebnerBasis +---------------+ | 3 21 11 4 | |x x - x x | | 1 3 2 4 | +---------------+ | 8 3 2 16 | |t x x - x x | | 3 2 4 1 3 | +---------------+ | 5 3 | |t x x - x x | | 3 1 3 2 4 | +---------------+ | 2 5 2 11 | |t x x - x x | | 3 2 4 1 3 | +---------------+ | 3 2 6 | |t x x - x | | 3 2 4 3 | +---------------+ | 4 | |t x - x x | | 3 1 2 3 | +---------------+ | 2 | |t x - x | | 2 3 2 | +---------------+ | 2 3| |t x x - t x x | | 2 2 4 3 1 3| +---------------+ | 3 4 | |t t x x - x | | 2 3 2 4 3 | +---------------+ | 2 | |t x x - t x x | | 2 2 4 3 1 3| +---------------+ | 2 3 2 | |t t x - x | | 2 3 4 3 | +---------------+ | 3 | |t x x - t x | | 2 3 4 3 1 | +---------------+ | 5 2 2 | |t t x - x x | | 2 3 4 1 3 | +---------------+ | 8 3 2 | |t t x - x | | 2 3 4 1 | +---------------+ |t x - t t x | | 1 3 2 3 4 | +---------------+ | 2 | |t x - t t x x | | 1 2 2 3 3 4| +---------------+ | 4 2 | |t x - t x | | 1 1 2 4 | +---------------+ | 2 | |t t t - x | | 1 2 3 3 | +---------------+ | 4 | |t t t x - x | | 1 2 3 4 1 | +---------------+ | 2 | |t t - x | | 1 3 4 | +---------------+

t_1^10*t_2^11*t_3^10 % Igb

2 t x x x 3 1 2 4 R

R = QQ[x_1..x_8,u_1,u_2, MonomialOrder=>{8,2}]
S = R[y,MonomialOrder=>Lex]
use R
I = ideal (2*x_1-u_1,2*x_2-u_2,2*x_3-u_1,2*x_4-u_2,x_5*u_1-x_6*u_2,x_5*u_2+x_6*u_1-u_1*u_2,
           (x_1-x_7)^2+x_8^2-x_7^2-(x_8-x_2)^2, (x_1-x_7)^2+x_8^2-(x_3-x_7)^2-(x_4-x_8)^2)
netList I_*
Ibar1 = substitute(I,S) + ideal (1-y*((x_1-x_7)^2+x_8^2-(x_5-x_7)^2-(x_6-x_8)^2))
Ibar1GB = gb Ibar1
netList entries transpose gens Ibar1GB

I = ideal (2*x_1-u_1,2*x_2-u_1-u_2,2*x_3-u_3,2*x_4-u_2,2*x_5-u_3,x_17-2*x_15,x_18-2*x_16,u_2+x_17-2*x_13,u_3+x_18-2*x_14,x_17-2*x_11+u_1,x_18-2*x_12,x_7*u_2-x_7*u_1+x_8*u_3,u_2*u_1-u_1*x_6,u_2*x_9-u_2*u_1+u_3*x_10,u_3*x_7-u_3*u_1+x_8*u_1-u_2*x_8,x_9*u_3-x_10*u_2,(x_19-x_4)^2+(x_20-x_5)^2-(x_19-x_1)^2-x_20^2,-(x_19-x_1)^2-x_20^2+(x_19-x_2)^2+(x_20-x_3)^3,-(x_19-x_1)^2-x_20^2+(x_19-x_15)^2+(x_20-x_16)^2,-(x_19-x_1)^2-x_20^2+(x_19-x_9)^2+(x_20-x_10)^2,-(x_19-x_1)^2-x_20^2+(x_19-x_13)^2+(x_20-x_14)^2,-(x_19-x_1)^2-x_20^2+(x_19-x_7)^2+(x_20-x_8)^2,-(x_19-x_1)^2-x_20^2+(x_19-x_15)^2+(x_20-x_16)^2

R = QQ[x_1..x_8,u_1,u_2, MonomialOrder=>{8,2}]
S = R[y,MonomialOrder=>Lex]
use R
I = ideal (2*x_1-u_1,2*x_2-u_2,2*x_3-u_1,2*x_4-u_2,x_5*u_1-x_6*u_2,x_5*u_2+x_6*u_1-u_1*u_2,
           (x_1-x_7)^2+x_8^2-x_7^2-(x_8-x_2)^2, (x_1-x_7)^2+x_8^2-(x_3-x_7)^2-(x_4-x_8)^2)
netList I_*
Ibar1 = substitute(I,S) + ideal (1-y*((x_1-x_7)^2+x_8^2-(x_5-x_7)^2-(x_6-x_8)^2))
Ibar1GB = gb Ibar1
netList entries transpose gens Ibar1GB