### Abstract

This entry formalizes the connection between Gröbner bases and
Macaulay matrices (sometimes also referred to as `generalized
Sylvester matrices'). In particular, it contains a method for
computing Gröbner bases, which proceeds by first constructing some
Macaulay matrix of the initial set of polynomials, then row-reducing
this matrix, and finally converting the result back into a set of
polynomials. The output is shown to be a Gröbner basis if the Macaulay
matrix constructed in the first step is sufficiently large. In order
to obtain concrete upper bounds on the size of the matrix (and hence
turn the method into an effectively executable algorithm), Dubé's
degree bounds on Gröbner bases are utilized; consequently, they are
also part of the formalization.

BSD License