Real radical initial ideals

We explore the consequences of an ideal I⊂R[x1,…,xn] 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 inw(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 w∈n(R>0), then the corresponding quadratic module is stable. In particular, if inw(I) is real radical for some w∈n(R>0) then ∑R2[x1,…,xn]+I is stable. This provides a method for checking the conditions for stability given by Powers and Scheiderer (2001) [PS].