Math 380, Spring 2018, Assignment 11
From cartan.math.umb.edu
Algebra begins with the unknown and ends with the unknowable.
- - Anonymous
Read:[edit]
- Section 3.1.
Carefully define the following terms, then give one example and one non-example of each:[edit]
- Projection map $\pi_l:\mathsf{k}^n\rightarrow\mathsf{k}^{n-l}$.
- Partial solution (of a system of polynomial equations).
Carefully state the following theorems (you do not need to prove them):[edit]
- Theorem concerning the computation of elimination ideals.
- Theorem relating $\pi_l(\mathbb{V}(I))$ to $\mathbb{V}(I_l)$.
- Example showing that equality may not hold in the previous theorem.
Solve the following problems:[edit]
- Section 3.1, problems 1 and 3.
- (Optional; you will want a computer algebra system) Section 3.1, problem 4.
- (Optional; you will want a computer algebra system) The "twisted cubic" is the locus of points $$C=\{(t, t^2, t^3)\,|\,t\in\mathbb{R}\}.$$ This is in fact an algebraic variety in $\mathbb{R}^3$, as you shall now show.
- (a) Using lex order in which $t>x>y>z$, compute a lexicographic Grőbner basis for the ideal $I=\left\langle x-t, y-t^2, z-t^3\right\rangle$.
- (b) Compute a basis for the first elimination ideal $I_1$.
- (c) Show that every point of $C$ belongs to $\mathbb{V}(I_1)$.
- (d) Show that every point of $\mathbb{V}(I_1)$ belongs to $C$.
- Thus, the equations corresponding to the generators you found in (b) are enough to "cut out" the curve $C$. The process you just went through is called "implicitizing" $C$; it is in some sense the reverse of "solving" a system of equations.
