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## Characteristics of the polynomial systems of equations

In general, the nonlinear systems of equations to be solved have the same number of equations as unknowns, and over-determined systems appear quite rarely in practice. However, it is important to mention that the way the system of equations is generated is the main reason why no over-determined systems appear. One example of an over-determined nonlinear system arises in the filter design procedure of the CCETT Company.

The number of equations (and of unknowns) goes from 16 in the case of the mathematical modelling of the mechanical components of a shock absorber from LABEIN, to the several thousands arising from some nonlinear systems coming from the application of Finite Elements to solve some differential or integral equations (forging process modellization from LABEIN and nuclear plant simulation from TECNATOM) or from the resolution of a variational problem (bridge design from APIA XXI).

The equations appearing in the nonlinear system are usually quite sparse, in many cases being presented in a non-expanded form which produces this sparsity. The degree is not usually very big: either the total degree or the degree in every unknown is bounded by two or three.

With respect to the coefficients, these were real numbers in every example except the EdF case where the coefficients belong to . In most cases they come from experimental data and thus are known only up to a limited accuracy. We also mention that, in the design of a variational CAD/CAE system at LABEIN, the coefficients are exact rational numbers.

Finally, parameters also appear quite often, but they are usually specialized in a first stage and, thus, the solution of the nonlinear system is obtained by interpolating the solutions of the specialized systems.

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