Abstract | We investigate the connection between Gröbner basis computation and Gaussian elimination. Our main goal is to construct for Gröbner
bases a concept similar to the one provided by subresultants for
polynomial remainder sequences.
Determinant polynomials are generalised for multivariate polynomials.
We associate to Gröbner bases computation Sylvester-like matrices
having the columns indexed by terms and containing on each row the
coefficients of a polynomial from the input basis, possibly shifted
by multiplication by a term. We show then that Gröbner basis polynomials computed during Buchberger algorithm can not, in general, be expressed as determinant polynomials of matrices of the form above. Even in simple examples where the input basis consists of three bivariate polynomials this is not always possible.
We consider then the more general problem of simulating the Gröbner
basis computation by Gaussian elimination. |