  By a result of Helton and McCullough, open bounded convex free semialgebraic
sets are exactly open (matricial) solution sets D_L of a linear matrix
inequality (LMI) L(X)>0. This paper gives a precise algebraic certificate for a
polynomial being nonnegative on a convex semialgebraic set intersect a variety,
a so-called "Perfect" Positivstellensatz.
  For example, given a generic convex free semialgebraic set D_L we determine
all "(strong sense) defining polynomials" p for D_L. This follows from our
general result for a given linear pencil L and a finite set I of rows of
polynomials. A matrix polynomial p is positive where L is positive and I
vanishes if and only if p has a weighted sum of squares representation module
the "L-real radical" of I. In such a representation the degrees of the
polynomials appearing depend in a very tame way only on the degree of p and the
degrees of the elements of I. Further, this paper gives an efficient algorithm
for computing the L-real radical of I.
  Our Positivstellensatz has a number of additional consequences which are
presented.
