Rational Complexity-One T-Varieties Are Well-Poised

Given an affine rational complexity-one T-variety X, we construct an explicit embedding of X in affine space Aⁿ. We show that this embedding is well-poised, that is, every initial ideal of I_X is a prime ideal, and we determine the tropicalization Trop(X). We then study valuations of the coordinate ring R_X of X which respect the torus action, showing that for full rank valuations, the natural generators of R_X form a Khovanskii basis. This allows us to determine Newton-Okounkov bodies of rational projective complexity-one T-varieties, partially recovering (and generalizing) results of Petersen. We apply our results to describe all integral special fibres of K* × T-equivariant degenerations of rational projective complexity-one T-varieties, generalizing a result of Süß and Ilten.