**Type Name:**`DPL`**Structure:**- for some .
**Description:**- A (possibly empty) list of monomials. The
empty list denotes the zero polynomial. The monomials in the list
are ordered according to the chosen ordering
`is_greater_mon`for monomials. All operations on polynomials preserve this ordering.The monomials are kept ordered since, in this case, the decomposition of a polynomial into the

*leading monomial*and its*remaining polynomial*, which is frequently needed during the Gröbner bases algorithm, can be implemented more efficiently.We also implemented the quotient of two polynomials, which is to be understood as ``exact division without remainder''. This is of course only possible if the divisor is invertible. In the case of polynomials over a field, which is the case we concentrated on, the invertible elements are the elements of the ground field (i.e. the constant polynomials). When attempting to divide by a non-constant polynomial, GRÖBNER will detect this and handle the error properly. In the case of polynomials over a domain that is no field, an error might be detected when trying to invert the divisor in the case where it is a constant that is not invertible. Implementing exact division in this fashion considerably simplifies the parser for input of polynomials.

Thu Sep 3 14:50:07 MDT 1998