Here is the problem statement:
Consider the polynomial ideal $I = \langle b-r_1-r_2, c-r_1r_2 \rangle \subset \mathbb{Q}[r_1,r_2,b,c].$ Show that $I \cap \mathbb{Q}[b,c] = \langle 0 \rangle$.
First, if you can solve this without the use of the theory of Gröbner bases, I would love to see a solution, but I am explicitly trying to solve this using some Gröbner basis.
Now, let me give my calculations, which do not yield the desired result. I will use the notation in Dummit and Foote. Put the lexicographic order $r_1 > r_2 > b > c$ on the polynomial ring and let $f_1 = -r_1-r_2 + b$ and $f_2 = -r_1r_2 + c$. Then, \begin{align*} S(f_1,f_2) &= \frac{r_1r_2}{-r_1}(-r_1-r_2 + b) - \frac{r_1r_2}{-r_1r_2} (-r_1r_2 + c)\\ &= -r_2(-r_1-r_2 + b) + (-r_1r_2 + c)\\ &=r_2^2-r_2b + c =: f_3. \end{align*} Next, \begin{align*} S(f_1,f_3) &= \frac{r_1r_2^2}{-r_1}(-r_1-r_2+b) - \frac{r_1r_2^2}{r_2^2}(r_2^2 - r_2b + c)\\ &=-r_2^2(-r_1-r_2 + b) - r_1(r_2^2-r_2b + c)\\ &=r_1r_2b-r_1c+r_2^3-r_2^2b =: f_4 &\\ & \\ S(f_2,f_3) &= \frac{r_1r_2^2}{-r_1r_2}(-r_1r_2 + c) - \frac{r_1r_2^2}{r_2^2}(r_2^2 - r_2b + c) \\ &=-r_2(-r_1r_2 + c)-r_1(r_2^2 - r_2b + c)\\ &=r_1r_2b - r_2c - r_1c =:f_5. \end{align*} Feeling like there was still hope this would work, I stopped with: \begin{align*} S(f_3,f_4) &= \frac{r_1r_2^2b}{r_2^2}(r_2^2 - r_2b+c) - \frac{r_1r_2^2b}{r_1r_2b}(r_1r_2 - r_1c + r_2^3 - r_2^2b)\\ &=r_1b(r_2^2-r_2b+c)-r_2(r_1r_2b-r_1c + r_2^3-r_2^2b)\\ &=-r_1r_2b -r_1r_2c + r_1bc -r_2^4 + r_2^3b =:f_6. \end{align*}
So, my question is: what's going on here? According to Mathematica, the Gröbner basis with this ordering is $ \lbrace c - b r_2 + r_2^2, -b + r_1 + r_2 \rbrace $ Notice the first element of this set if what I call $f_3$ and the second element is $f_1$. Thus, I would think my computation should have terminated at $S(f_1,f_3)$ and $S(f_2,f_3)$.
I suspect I am making an arithmetic error, but by now I have some pretty bad tunnel vision so I can't spot it. Of course, I could also be making a more serious error somewhere, e.g., taking the wrong least common multiple.
Any help is appreciated.