-- Examples with singular points. R = QQ[x,y, MonomialOrder=>Lex]; f = y^2-x^3-x^2; -- Here is how to compute the partial derivative withs respect to x and y diff(x,f) diff(y,f) -- -- Here we find the singular points of a curve given by -- f = 4*(x^2+y^2)^4-16*x^2*y^2+1 -- R = QQ[x,y, MonomialOrder=>Lex]; f = 4*(x^2+y^2)^4-16*x^2*y^2+1; I = ideal(f, diff(x,f), diff(y,f)); G = (entries(gens(gb(I))))_0 -- (We finish the problem by hand.) -- -- Here we find the singular points of a parametrized curve -- x = t^2-1 -- y = t^3+t-1 -- -- First, we find the implicitization. R = QQ[t,x,y, MonomialOrder=>Lex]; I = ideal( x- (t^2-1), y-(t^3+t-1) ); G = (entries(gens(gb(I))))_0 -- Only the 0th entry contains no t variables, so the implicit equation is: f = G#0 -- We construct the ideal of singular points: I2 = ideal(f, diff(x,f), diff(y,f)) G2 = (entries(gens(gb(I2))))_0 -- So the only singular point is (-2,-1) -- -- -- -- -- Here I sketch how to solve Question 7 and 8 from PS3 -- (This has nothing to do with singular points) -- --Q7 -- a=cos(t) -- b=sin(t) -- c=cos(u) -- d=sin(u) R=QQ[a,b,c,d,x,y,z,MonomialOrder=>Lex]; I = ideal(x-(2+a)*c, y-(2+a)*d, z-b, a^2+b^2-1, c^2+d^2-1); G = (entries gens gb I)_0 --Q8 R=QQ[a,b,c]; I = ideal(a+b+c-3, a^2+b^2+c^2-5, a^3+b^3+c^3-7); (a^4+b^4+c^4) % I (a^5+b^5+c^5) % I (a^6+b^6+c^6) % I