Singular

Hello,

I cant' understand difference between groebner and standard basises.
If I'm not mistaken, the definition from 'std' manual "A standard basis is a set of generators such that the leading terms generate the leading ideal" is the same as Groebner basis definition in Cox, Little, and O'Shea.
Can anyone help me? If you know some comprehensive definitions, please give me links to them.

There are several commands to compute a standard basis: std, slimgb, stdfglm, etc... which all use one specific algorithm.
groebner uses some heuristic to find which of the commands to call
to compute a standard basis.

Conclusion: if you know which algorithm works best in your case,
use std, slimgb or whatever.
If not, rely on the heuristic and use groebner.

Thanks,
but does this mean that groebner basis and standard basis are the same things?

Simply said, standard bases and Groebner bases are the same thing if the monomial ordering is global.

Although the definition of a standard basis and a Groebner basis is formally the same, the notion standard basis is used in a more general context, namely for arbitrary monomial orderings (including global, local and mixed orderings) while the notion Groebner basis is used only for global orderings (equivalent to well-orderings). Since most authors treat only well-orderings, the difference is not visible.

In contrast to the definition, it is important to note that a different algorithm is needed to compute standard bases for non-global orderings, since Buchberger's algorithm does not terminate.

In the Singular system however the commands 'std' and 'groebner' can be used for both, global and non-global orderings ('groebner' is a library command which tries to find heuristically the best strategy and simply applies std if the ordering is not global).

Note that results of std computations with local and mixed orderings have to be interpreted in the localization of a polynomial ring at the maximal ideal at 0. They are necessary e.g. for computational local algebraic geometry.

Arbitrary monomial orderings (not necessarily well-orderings) have been first introduced and implemented by the Singular team since the 1990's. The general theory is due to the authors of the textbook (where you find the theoretical background together with many examples):

G.-M. Greuel, G. Pfister: A SINGULAR Introduction to Commutative Algebra, Springer Verlag 2002 (2nd Edition 2007).

Thanks a lot.
It was very helpful.