k = QQ n = 4 R = k[x_1..x_n] describe R -- SETTING A MONOMIAL ORDER Rlex = k[x_1..x_n, MonomialOrder=>Lex] Relim = k[x_1..x_n, MonomialOrder=>Eliminate 1] -- product order R121 = k[x_1..x_n, MonomialOrder=>{1,2,1}] describe oo Rlexlex = k[x_1..x_n, MonomialOrder=>{Lex=>2,Lex=>2}] describe oo Rweight = k[x_1..x_n, Weights=>{3,1,0,0}] describe oo -- NORMAL FORM use R I = ideal(x_1^3 - 2*x_1*x_2, x_1^2*x_2 - 2*x_2^2 + x_1, x_3^2, x_3-x_4^3); (x_1^3 - 2*x_1) % I x_1^3 % I (x_4^4) % I use Rlex -- execute 4 previous lines and compare with NFs you got for R -- GROEBNER BASES gb I gens gb I entries gens gb I entries gens gb sub(I,R) entries gens gb sub(I,Relim) -- gb under the hood radical I isSubset(I, radical I) isSubset(ideal x_2, I) isSubset(ideal x_2^2, I) isSubset(ideal x_2^3, I) -- SYZYGIES F = gens I S = syz F F*S use R -- syzygies of columns of a matrix M = transpose matrix subsets(R_*,2) syz M -- ELIMINATION R = Rlex M = random(R^4,R^5) a = transpose (vars R | matrix{{1}}) Ma = M*a I = ideal Ma gens gb I I2 = ideal apply(4,i->random(2,R)+i^2) gens gb I2 eliminate({x_1,x_2,x_3},I2) -- Pythagorean theorem via implicitization clearAll R = QQ[x,y,a,b,c] I = ideal {a*b-(c-x)*y-x*y, a*y-(c-x)*b, a*x-y*b} eliminate({x,y},I) factor oo_0