Dmitrii Pasechnik

Activity: Hosting a visitorAcademic

Description

Title: An efficient sum of squares certificate for 4-ary 4-ic

Abstract: Hilbert's 17th problem (resolved by Artin and Schreier) can be
formulated as the existence, for any globally nonnegative n-ary
homogeneous polynomial (a.k.a. form) f, of a sum of squares (s.o.s., for
short) form q so that fq is a sum of squares (thus f is nonnegative).
Note that once the degree of q is known, finding it can be done by
solving a semidefinite optimisation (SDP) feasibility problem.

For n=3, Hilbert has shown a quadratic, in the degree of f, bound on the
degree of q. In general, the best known degree bounds are huge. The next
interesting case is n=4, and f of degree 4. We show that in this case
there exists a product of two non-negative quadrics q so that qf is an
s.o.s. of quartics. As a step towards deciding whether it is sufficient
to use a quadratic multiplier q, we show that there exist non-s.o.s.
non-negative 3-ary sextics ac−b2, with a, b, c of degrees 2, 3, 4,
respectively (this gives a new class of nonnegative 3-ary sextics which
are not s.o.s.).
Period9 Mar 2020
Visiting fromPembrooke College, University of Oxford (United Kingdom)
Visitor degreePhD
Degree of RecognitionLocal