Math 380, Spring 2018, Assignment 6

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.

- Goethe

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

1. Ideal membership test for univariate polynomials (i.e. $f\in\left\langle g\right\rangle$ if and only if...).
2. Example demonstrating the failure of the corresponding test for multivariate polynomials.

Solve the following problems:

1. Section 2.3, problem 6 (Hint: try to find a "syzygy" of $f_1$ and $f_2$, i.e. a non-linear combination $Af_1+Bf_2$ in which there is a cancellation of leading terms.)
2. Section 2.3, problem 7 (use the same hint as above).

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