Math 380, Spring 2018, Assignment 11

Algebra begins with the unknown and ends with the unknowable.

- Anonymous

Read:

1. Section 3.1.

Carefully define the following terms, then give one example and one non-example of each:

1. Projection map $\pi_l:\mathsf{k}^n\rightarrow\mathsf{k}^{n-l}$.
2. Partial solution (of a system of polynomial equations).

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning the computation of elimination ideals.
2. Theorem relating $\pi_l(\mathbb{V}(I))$ to $\mathbb{V}(I_l)$.
3. Example showing that equality may not hold in the previous theorem.

Solve the following problems:

1. Section 3.1, problems 1 and 3.
2. (Optional; you will want a computer algebra system) Section 3.1, problem 4.
3. (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.

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