  A bad point of a positive semidefinite real polynomial f is a point at which
a pole appears in all expressions of f as a sum of squares of rational
functions. We show that quartic polynomials in three variables never have bad
points. We give examples of positive semidefinite polynomials with a bad point
at the origin, that are nevertheless sums of squares of formal power series,
answering a question of Brumfiel. We also give an example of a positive
semidefinite polynomial in three variables with a complex bad point that is not
real, answering a question of Scheiderer.
