This subsection is devoted to collect some of the concrete needs explicitly stated in the answers. They are very different in type and content due to their strong dependence on the concrete area/problem they come from.

One of the most common request from companies, enterprises and R&D
laboratories, apart from speed and accuracy, is the need of
interactivity for the user and interconnectivity in the manipulation of
different mathematical software packages. The provision of good-quality
interfaces, together
with the possibility of exporting functions to be integrated easily in
specialized `C++` libraries, would greatly simplify the task of
researchers in industry.

The availability of up-to-date methods for polynomial system solving would be very useful. For example, a nonlinear system of equations containing parameters is only solved by specializing the parameters several times and solving the corresponding specialized system, while some parametric computations (Gröbner Basis, resultants, etc) could be useful. Another example of this situation appears when convergence problems arise in the application of a Newton-like method: instead of replacing the algorithm by another one which is more powerful or better adapted (symbolic method, homotopy, ), the mathematical model is simplified or changed.

A particular problem from a fluid structure interaction at LABEIN shows how the FRISCO capabilities can be useful in the near-future. The problem is no more than the optimization of an objective function whose analytical expression is not available and whose value is obtained when a concrete system of polynomial equations is solved (the input values of the function provide the coefficients of the system). This implies that an analytical expression of the function could either be computed exactly, or approximated by performing an elimination procedure.

Finally, and related to the previous point, a good collection of test
suites, not only with statements of the problems and results,
but also containing calling
sequences allowing the user to experiment with every system on a WWW server,
would be really useful. A preliminary version of these WWW servere can be found at
`http://www.inria.fr/safir/POL/`

and `http://janet.dm.unipi.it/posso_demo.html`