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Reduction of polynomial systems

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.

Jan Verschelde