Straight-line Programs in Polynomial Equation Solving

Solving symbolically polynomial equation systems when intermediate and final polynomials are represented in the usual dense encoding turns out to be very inefficient: the sizes of the systems one can deal with do not respond to realistic needs. Evaluation representations appeared in this frame a decade ago as a new possibility to treat new families of problems. We present a survey of the most recent complexity results for different polynomial problems when polynomials are encoded by evaluation (straight-line) programs. We also show surprising mathematical by-products, such as new mathematical invariants and results, that appeared as a consequence of the search of good algorithms.