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Real radical initial ideals

We explore the consequences of an ideal I of real polynomials having a real radical initial ideal, both for the geometry of the real variety of I and as an application to sums of squares representations of polynomials. We show that if in_w(I) is real radical for a vector w in the tropical variety, then w is in the logarithmic set of the real variety. We also give algebraic sufficient conditions for w to be in the logarithmic limit set of a more general semialgebraic set. If in addition the entries of w are positive, then the corresponding quadratic module is stable. In particular, if in_w(I) is real radical for some positive vector w then the set of sums of squares modulo I is stable. This provides a method for checking the conditions for stability given by Powers and Scheiderer.