-- This is code related to lectures from Feb 1 -- Math 615, Winter 2013 -- By: Daniel Erman -- Basics -- If you ask M2 to compute a Grobner basis, it will automatically compute the reduced GB, -- and sort the basis elements by their leading terms R = QQ[x,y, MonomialOrder => Lex] viewHelp MonomialOrder I = ideal(x^3,x^2*y+y^3) gb I -- The command "gb" will compute and store the Grobner basis, but it will not display it! -- To see the actual Grobner basis, try: gens gb I -- Reduced GB Example from lecture R = QQ[x,y] -- Not specifying a monomial order means we use GRevLex I = ideal(x^3+x*y^2, x*y+y^2) gens gb I -- Ideal membership example -- we continue with the above notation. f = x^2*y^3 f % I -- this means that there has to be a way to write f = a*x^3+b*(x^2*y+y^3). What are a & b? f // gens I -- So this matrix provides a "witness" to the fact that f lies in I, namely: M = f // gens I; (gens I)*M == f -- Elimination of variables examples R = QQ[x,a,b,c, MonomialOrder => Lex] f = a*x^2+b*x+c; I=ideal(f,diff(x,f)) gens gb I R = QQ[x,y,a,b, MonomialOrder => Lex] f = y^2-x^3-a*x-b; I=ideal(f,diff(y,f),diff(x,f)) gens gb I -- Here is an example that I didn't do in class. This involves projecting an elliptic -- curve V(I) in PP^3 down to PP^2 via the map [x:y:z:w] --> [y:z:w]. R = QQ[x,y,z,w, MonomialOrder=> Lex] I = ideal(x*w-y*z,x*y+z^2+w^2) dim (minors(2,jacobian I) + I) degree I gens gb I -- the first entry of the above matrix is the defining equation of the curve inside of PP^2. (gens gb I)_(0,0) -- Solving polynomial equations R = QQ[x,y, MonomialOrder=> Lex] I = ideal(x^2+2*y^2-3,x^2+x*y+y^2-3) degree I gens gb I -- so y^3-y=0. Thus y=0,1, or -1. -- Now substituting these values for y and solving yields the remaining solutions. -- Implicitization -- This is the implicitization example from class, where we compute the image of a -- map AA^2-->AA^3. R = QQ[s,t,x,y,z, MonomialOrder=> Lex] I = ideal(x-s^2-t^2,y-s^3-t^3,z-s*t) (gens gb I)_(0,0) -- Radical membership -- Here is a radical membership example R = QQ[x,y] I = ideal(x^3-x^2*y, x*y^2+y^3) -- Let's show that x belongs to the radical of I. S = R[t] J = I+ideal(1-t*x) gens gb J -- By Rabinovitch's trick, we see that x lies in the radical of I.