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 a
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Time is passing by, and work and study go ahead. In particular I am now studying different kind of boundary conditions that are necessary and/or sufficient to solve problems of constrained polynomial optimization. We are proving that different conditions present in the literature are more or less equivalent, and that under these hypothesis the moment relaxations enjoy good properties of convergence. I also planned with my supervisor four meetings/conferences between January and the end of April: of course the two POEMA ones, but also two more: la journée Moments and Francophone Computer Algebra Days - Journées nationales de calcul formel : they will be busy, interesting months.
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This is part of the POEMA Blogs that the ESRs of the project will maintain for the next three years. I started working at INRIA (Sophia Antipolis) in October. The research center is really a good place where to study: the team AROMATH, where I am hosted, is very international and there are often visitors from outside coming and giving talks, the offices are great (I will share mine with Tobias Christopher - ESR 10), and also the whether is good. I spent first weeks reading articles and discussing with my supervisor: the idea is to use techniques from commutative algebra, real algebraic geometry and moments to address questions in polynomial optimization. In particular now we are trying to prove (and improve) results from the sum of square relaxation to the moment relaxation.