-- compute the image of a morphism phi(x) = (x^2, x^3) R = QQ[x,y1,y2,MonomialOrder=>Lex]; I = ideal(y1-x^2,y2-x^3) gens gb I eliminate(I, x) -- compute the image of a circle under a rational map -- rho = (x1/(1-x2), (1+x2)/(1-x2)) R = QQ[x1,x2,z,y1,y2,MonomialOrder=>Lex]; I = ideal(x1^2 + x2^2-1); J1 = ideal(y1 - x1*z, y2 - (1+x2)*z, z*(1-x2)-1) + I; J2 = ideal(y1 - x1*z*(1-x2), y2 - (1+x2)*(1-x2)*z, z*(1-x2)^2-1) + I; J1==J2 I1 = eliminate(J1, {x1,x2,z}) I2 = eliminate(J2, {x1,x2,z}) I1 == I2 -- compute the image of a rational map -- rho(x) = ((x^2-1)/(x^2 + 1), (2*x)/(x^2 + 1)) R = QQ[x,z,y1,y2,MonomialOrder=>Lex]; I = ideal(z*(x^2+1)-1, y1 - (x^2-1)*z, y2 - 2*x*z) gens gb I eliminate(I, {x,z}) -- finding a Groebner basis for polynomials R = QQ[x,y,MonomialOrder=>GRevLex]; I = ideal(x*y-1, x^2+y^2-2); gens gb I leadTerm(x*y-1) leadTerm(I) -- finding a Groebner basis for polynomials from class 1 R = QQ[x,y]; I = ideal(x^3-x^2+y^2, x-2, y^2-1); gens gb I -- Calculating leading term ideal, normal form, standard monomials R = QQ[x,y,MonomialOrder=>Lex]; I = ideal(x^2-1,y^2-1); leadTerm(I) gens gb I --normal forms (x^2*y+x*y^2) % I (x^2*y - x^2 + y^2 -y) % I (x-y^2)% I -- standard monomials S = R/I basis S -- Calculating syzygies R = QQ[z,y,x, MonomialOrder=>Lex] f1 = z - x^3; f2 = y - x^2; g = x^3*z -y^3; m = matrix{{f1,f2,g}}; S = syz m m*S_{1} -- primary Decomposition R = QQ[x,y]; I = ideal(x^2, x*y); primaryDecomposition I associatedPrimes I decompose I minimalPrimes I I:ideal(x) I:ideal(y) -- singular point of a curve R = QQ[x,y]; f = x^3-x^2+y^2; I = ideal(f,diff(x,f), diff(y,f)); J = ideal(x,y); I:J restart -- Hassett Ex 8.18: not every power of a prime ideal is primary R = QQ[a,b,c,d,e,f]; M = matrix({{a,b,c},{b,d,e},{c,e,f}}); I = minors(2,M) J = I*I isPrime I isPrimary J primaryDecomposition J associatedPrimes J minimalPrimes J L = associatedPrimes J; L_(0) == I