Certificates of Algebraic Positivity
Frank Sottile
Texas A&M University

  Positivity is a distinguishing property of the field of real numbers.  Writing
a  polynomial as  a sum  of squares  gives a  certificate that  it  is positive.
Hilbert showed that a positive homogeneous quartic polynomial in three variables
(ternary quartic) is  a sum of three squares of  quadratic polynomials.  He also
showed that  there are  positive polynomials of  every higher degree  or greater
number of  variables with no such  sum of squares representations.   This led to
his  17th problem---to  determine  whether a  positive  polynomial is  a sum  of
squares of rational functions.  This was answered in the affirmative by Artin in
1926.

  Recently, positive  polynomials have  have undergone a revival.  In the 1990's
Lasserre realized  that recent theoretical results from  real algebraic geometry
and semi-definite programming could be combined to give effective algorithms for
solving a  class of relaxations  of hard optimization problems.   The relaxation
replaces positivity by sum of squares representation.

  I will  briefly survey  the history of  positive polynomials and  these modern
applications, and  then discuss a  recent strengthening of Hilbert's  Theorem on
ternary quartics:  a positive ternary quartic is  a sum of squares  in exactly 8
inequivalent ways.