I am studying something that touches on Groebner algorithms at the moment and it seems like i am missing something obvious about the relationship between three definitions that feel like they should be related.
1) A set of polynomials $\{P_1,\dots,P_n\}$ is said to be algebraically independent if the Jacobian of the vector $\mathbf{P}=(P_1,\dots,P_n)$ is non singular.
2) A set of polynomials $\{P_1,\dots,P_n\}$ are the generating set for a polynomial ideal $I(\mathbf{x})$ if it is the smallest ideal containing them.
3) A Groebner base $G$ for a polynomial ideal $I(x)$ if it is the generating set for $I(x)$ and there are no nonzero polynomials in $I(x)$ of lower degree than those in the Groebner base.
From computationally using Groebner Bases with maple I know that I expect them to be sets of polynomials that are algebraically independent - but I haven't seen this written anywhere and i don't think it's implied by definitions 2 or 3. Are Groebner bases also sets of algebraically independent polynomials? And if so how do you get from definitions 2 and 3 to that?
Also I have seen the phrases Groebner basis, and Groebner base both used in different books/websites. Is there a difference between the two terms? Or could one be a typo?