  Recent breakthroughs have been made in the use of semidefinite programming
and its application to real polynomial solving. For example, the real radical
of a zero dimensional ideal, can be determined by such approaches as shown by
Lasserre and collaborators. Some progress has been made on the determination of
the real radical in positive dimension by Ma, Wang and Zhi. Such work involves
the determination of maximal rank semidefinite moment matrices. Existing
methods are computationally expensive and have poorer accuracy on larger
examples.
  This paper is motivated by problems in the numerical computation of the real
radical ideal in the general positive case.
  In this paper we give a method to compute the generators of the real radical
for any given degree $d$. We combine the use of moment matrices and techniques
from SDP optimization: facial reduction first developed by Borwein and
Wolkowicz. In use of the semidefinite moment matrices to compute the real
radical, the maximum rank property is very key, and with facial reduction, it
can be guaranteed with very high accuracy. Our algorithm can be used to test
the real radical membership of a given polynomial. In a special situation, we
can determine the real radical ideal in the positive dimensional case.
