All problems are from "Using algebraic geometry" unless stated otherwise.
Try to look at all exercises. Write down the solutions to at least 3 problems by the due date.
You are encouraged to turn in the hardest (in your opinion) problems that you can solve. Feel free to work in groups, but make sure you understand the solutions that you write down.
[due Apr 28] exercises in the notes on Numerical Algebraic Geometry.
[due Mar 13] (4.1) 6, 11;
(4.2) 2, 4, 8, 15;
(4.3) 5, 12;
(4.4) 1, 3;
(4.5) 3, 4, 6.
- Let $R=k[x]$, $M = \langle x \rangle$. Show that every ideal of the local ring $R_M$ is a power of the maximal ideal $MR_M$.
- Let $R=k[x,y]$, $M = \langle x,y \rangle$. Give an example of a 0-dimensional ideal $I$ of $R$ such that its extension $IR_M$ is not a monomial ideal (i.e., can't be generated by cosets of monomials of $R$).
[due Feb 25] (2.1) 7, 10;
(E1) Show that any variety is a projection of a variety defined by polynomials of degree at most 2.
(2.2) 6, 12, 18;
(2.3) 7, 10;
(E2) Let $>_1$ and $>_2$ be two graded monomial orders on $k[x,y]$. Construct a conversion algorithm that given a Gröbner basis with respect to $>_1$ returns a Gröbner basis for $\langle G \rangle$ with respect to $>_2$. [The ideal generated by $G$ is not assumed to be 0-dimensional.]
(2.4) 9, 12;
Consider the companion matrix $A_f$ of a univariate polynomial $f$ (see page 83).
- Prove that the characteristic polynomial of $A_f$ is $f$.
- Prove that the minimal polynomial of $A_f$ is $f$.
- Find an eigenvector of $A_f$ corresponding to the root $a$ of $f$.
- Find generalized eigenvectors of $A_f$ corresponding to the root $a$ of $f$ of multiplicity $m$.
[due Jan 23] (Section 1.1) 5-13; (1.2) 5, 7-9; (1.3) 2, 5, 7, 9, 10; (1.4) 4, 8-10.