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.