Gaussian elimination of a system of linear equations is another special case where the degree of all polynomials equals one.
A crude version of this algorithm to find a basis for an ideal I of a polynomial ring R proceeds as follows: The polynomial Sij is commonly referred to as the S-polynomial, where S refers to subtraction (Buchberger) or syzygy (others).
The algorithm terminates because it is consistently increasing the size of the monomial ideal generated by the leading terms of our set F, and Dickson's lemma (or the Hilbert basis theorem) guarantees that any such ascending chain must eventually become constant.
This allows, in theory, to use linear algebra over the vector space of the polynomials of degree bounded by this value, for getting an algorithm of complexity
On the other hand, there are examples[2] where the Gröbner basis contains elements of degree and the above upper bound of complexity is optimal.