I have a upcoming test tomorrow.
In my class, my professor didn't give us complete proofs but taught us how to compute Gröbner base and he told me computing problems are gonna be on exam. I hate this kind of teaching, but it's an exam so I have to stupidly memorize how to compute without knowing what I'm doing.
Here are few questions I'm confused about:
What is the difference bettwen LP & LT ?
What is $S(g_i,g_j)$?
Does this mean, for example if $g_1=x+y$ and $g_2=y^2$ where $x<y$, then $S(g_1,g_2)=xy$?
Here is a theorem I should memorize
Let $\{g_1,g_2,...,g_n\}$ be a base for $I$. Then $\{g_1,...,g_n\}$ is a Gröbner base iff $S(g_i,g_j)$ can be reduced to 0 by repeatedly dividing remainders by elements of $G$.
What does this theorem mean? and How do I determine whether it is a Gröbner base?
I hope please someone help me with at least one example for each question..
Thank you in advance