Sums of squares and applications

A real polynomial that can be written as a sum of squares of polynomials
has non-negative values everywhere, while the converse is usually far
from being true. Studying the gap between non-negative polynomials and
sums of squares, in generalized settings, is a guiding principle for
these lectures. I will start with a series of classical results by
Hilbert and Artin, and then turn to a series of Positivstellensätze in
various variants. Those are more recent and have a variety of
applications, of which a few will be sketched. In particular, applications
to polynomial optimization have become very important in recent years,
and I'll discuss them in some detail.


Two very good references:

M. Marshall:
Positive Polynomials and Sums of Squares.
Mathematical Surveys and Monographs vol 146, AMS, 2008

T. Netzer:
Real Algebra and Geometry.
Script, Innsbruck University, available at


M. Knebusch, C. Scheiderer:
Real Algebra - A First Course.
Translated from the 1990 German original.
Springer Nature, 2022 (to appear)


Claus Scheiderer

Published Apr. 25, 2022 8:24 AM - Last modified June 9, 2022 5:47 PM