Block 1 due to 18.10.: prove the two statements out of the "dictionary" in affine.mws, lines 6 and 9 Block 2 due to 25.10.: write a program for deciding equality of ideals given by finite sets of generators (I apologize for this exercise being too easy, as I did not advance in the course to the next "real" exercise!) Block 3 due to 22.11.: prove the Jacobian conjecture for n=2 and at most quadratic polynomials (exercise 1 in affine.mws) Some hints: - use "subs" with a set of substitutions for composition of polynomial maps - to find an inverse, try "solve({F=u,G=v},{x,y})" - for polynomial maps without constant part (translations), the linear parts compose just like matrix multiplication. This knowledge can be used to simplify the linear part to the unity matrix Block 4 due to 29.11.: (exercise 2 in affine.mws) The normalization as computed by the Maple command integral_basis seems to be depend on the ordering of the variables. But the theory tells that the ordering of variables does not matter. Compare the results of integral_basis(y^2-x^3,y,x) and integral_basis(y^2-x^3,x,y) and show that they are essentially the same. Formulate a more precise statement: what means "essentially"? Block 5 due to 13.12.: (1) compute the singularities of the space curve with equations y^2-xz+y=x^2y+y-z^2+1,x^3-yz+x using the Jacobian criterion (2) show by dimension count that there are cubic curves without rational parametrization, using the following observations: - the implicit equation can be obtained from the parametrization (x=p(t)/r(t),y=q(t)/r(t)) by Resultant(p(t)-xr(t),q(t)-yr(t),t) This resultant is nonzero unless p,q,r have a common divisor - the implicit equation has degree >3 if at least one of p,q,r has degree >3 - a reparametrization of the form t->(at+b)/(ct+d), where ad-bc not equal 0, leads to the same curve The proof does not have to be complete, in particular you may neglect lower-dimensional cases. Block 6 due to 17.1.: prove: for n>=3, the generic plane curve of degree n has inflection tangents Block 7 due to 24.1.: give an example of a statement involving generic points and prove it using the algebraic concept of generic points Block 8 due to 31.1.: [proof of a statement given in the lecture] Block 9 due to 9.3.: study the 3rd section of the worksheet projective.mws and do the exercise, i.e. write a program for computing the projective normalization