In these months I had the opportunity to meet and discuss with the ESR's and other PhD and researchers at the POEMA workshop and in INRIA Saclay. It is a great advantage the opportunity to share ideas and have suggestions from all them. In particular I would like to thank Monique Laurent and Mohab Safey El Din for their counterexamples and advice.

The work is going on, and in particular now the Moment Approach is playing a decisive role. An explicit description of the properties of (a section of the cone of) the positive functionals has given new light on some previous facts we discovered, clarifying statements and proofs. Our first results are now ready to be presented, and I will make a presentation of them at the Francophone Computer Algebra Days. Here you can find the abstract.

Up to now we can prove exactness in some cases (finite feasible set, gradient variety, isolated singularities). The future work will focus on generalizing these results, and to study the geometry and the dependence from the presentation of the positive functionals we use in our proofs.

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