The goal of reduction is to rewrite the system into an equivalent one (i.e.: with the same finite solutions) that has a lower total degree, so that fewer solution paths need to be followed. Sometimes reduction can already detect whether a system has no solutions or an infinite number of solutions.

**Linear Reduction**- performs row-reduction on the coefficient matrix of the system.
**Sparse Linear Reduction**- brings the coefficient matrix of the system in a diagonal format.
**Nonlinear Reduction**- replaces polynomials by Subtraction-polynomials to eliminate highest-degree monomials. This type of reduction is more powerful, but also more expensive. Bounds have to be set to limit the combinatorial enumeration.