  For information-theoretic quantities with an asymptotic operational
characterization, the question arises whether an alternative single-shot
characterization exists, possibly including an optimization over an ancilla
system. If the expressions are algebraic and the ancilla is finite, this leads
to semialgebraic level sets. In this work, we provide a criterion for
disproving that a set is semialgebraic based on an analytic continuation of the
Gauss map. Applied to the von Neumann entropy, this shows that its level sets
are nowhere semialgebraic in dimension d>2, ruling out algebraic single-shot
characterizations with finite ancilla (e.g., via catalytic transformations). We
show similar results for related quantities, including the relative entropy,
and discuss under which conditions entropy values are transcendental,
algebraic, or rational.
