I am wondering if some polynomials are given, how do we know whether they form Groebner basis or not. Note that it is not necessary that given poly's form a reduced Groebner basis.
I know how to find a Grobner basis of an ideal but Grobner basis is not unique. So I don't know how to determine if they are Groebner basis.
You can use a variant of Buchberger's algorithm to check if a given set of generators is a Gröbner basis. Specifically:
For each pair of generators, compute the $S$-polynomial.
Use the multivariate division algorithm to reduce the $S$-polynomial with respect to the generators as much as possible.
If the remainder is zero for every possible $S$-polynomial, then the generating set is a Gröbner basis.