Santiago Laplagne, University of Buenos Aires: Exact polynomial sum of squares decompositions
Referent: Santiago Laplagne, Exact polynomial sum of squares decompositions
Mittwoch, 12.06.2019, 16:15 h
Ort:Raum 418-436
Writing a positive polynomial as a sum of squares (SOS) is an important problem in computational mathematics, with many applications in continuous and combinatorial optimization. A natural question is to which extent can an approximated sum of squares decomposition be rounded off to an exact rational decomposition, so a to provide a non-negativity certificate. In this talk we first prove that if a rational polynomial is the sum of two squares in an algebraic extension of odd degree of the rational numbers, then it can always be decomposed as a rational SOS. For the case of more than two polynomials we provide an explicit example of a rational polynomial that is the sum of three squares with coefficients in Q(alpha), alpha the cubic root of 2, that cannot be decomposed as a rational SOS. We will show computations in Maple and Singular to answer these questions and find the decompositions.