Séminaires : Séminaire des Thésards

Le séminaire des thésards est l'occasion pour les doctorants de présenter des résultats et des problématiques dignes d'intérêt devant un public de non-spécialistes. L'ambiance y est informelle ; poser des questions naïves est encouragé, et les questions moins naïves sont bienvenues dans la mesure où elles n'entravent pas le bon déroulement de l'exposé.

Un jeudi sur deux à 18h00, en alternance entre Jussieu et Sophie Germain.

Pick a polynomial of degree d in n variables which takes only non-​negative values. Is it the sum of squares of polynomials? The answer was given by Hilbert in 1888 who showed that this is only true in three cases: n=1 and d arbitrary, n arbitrary and d=2, n=2 and d=4. We will prove the first two cases of this theorem and give Motzkin’s explicit counterexample of a non-​negative polynomial which is not a sum of squares of polynomials. A refinement of the above question leads to what is now known as Hilbert’s 17th problem: Is every non-​negative polynomial a sum of squares of rational functions? The answer to this question is yes, and was given by Artin in 1927. Its proof was very influential and fostered the development of real algebra. We will discuss the relevant concepts from real algebraic geometry, and finish by sketching a proof of Artin’s theorem.