I want to find the number of solutions to a Grobner basis.
For example: $G = \{x^2-1, y^2 -1 , (x-1)(y-1)\}$.
In the textbook, it says there are three irreducible monomials $\{ 1, x,y\}$ and hence three solutions.
I don't understand how to get these irreducible monomials from $G$.
For an ideal $I$, $\Delta_<(I)$,the footprint of $I$ is defined to be the set of monomials $X$ such that $X$ is not a leading monomial of any polynomial in $I$ ($<$ denotes the fixed monomial ordering). Here, $G$ is a Groebner Basis thus it is enough to check the monomials which is not divisible by $x^2,y^2$ or $xy$ which is the set $\Delta_<(I)=\{1,x,y\}$. If the footprint of an ideal is finite then the number of elements in the footprint is equal to the number of points in the zero locus, counted with multiplicity.