On semidefinite-representable sets over valued fields
For researchers in optimization and algebraic geometry, this work generalizes foundational concepts to valued fields, but the results are theoretical and incremental in nature.
This paper extends the theory of semidefinite-representable sets from the real numbers to valued fields, proving that key properties carry over and providing examples of non-polyhedral spectrahedra and sets that are semidefinite-representable but not spectrahedra over such fields.
Polyhedra and spectrahedra over the real numbers, or more generally their images under linear maps, are respectively the feasible sets of linear and semidefinite programming, and form the family of semidefinite-representable sets. This paper studies analogues of these sets, as well as the associated optimization problems, when the data are taken over a valued field $K$. For $K$-polyhedra and linear programming over $K$ we present an algorithm based on the computation of Smith normal forms. We prove that fundamental properties of semidefinite-representable sets extend to the valued setting. In particular, we exhibit examples of non-polyhedral $K$-spectrahedra, as well as sets that are semidefinite-representable over $K$ but are not $K$-spectrahedra.