/usr/include/polymake/next/ is in libpolymake-dev-common 3.2r2-3.
This file is owned by root:root, with mode 0o755.
..
/usr/include/polymake/next/AccurateFloat.h
/usr/include/polymake/next/Array.h
/usr/include/polymake/next/Bitset.h
/usr/include/polymake/next/FacetList.h
/usr/include/polymake/next/GenericStruct.h
/usr/include/polymake/next/Graph.h
/usr/include/polymake/next/IncidenceMatrix.h
/usr/include/polymake/next/Integer.h
/usr/include/polymake/next/Map.h
/usr/include/polymake/next/Matrix.h
/usr/include/polymake/next/Plucker.h
/usr/include/polymake/next/Polynomial.h
/usr/include/polymake/next/PowerSet.h
/usr/include/polymake/next/PuiseuxFraction.h
/usr/include/polymake/next/QuadraticExtension.h
/usr/include/polymake/next/Rational.h
/usr/include/polymake/next/RationalFunction.h
/usr/include/polymake/next/Set.h
/usr/include/polymake/next/Smith_normal_form.h
/usr/include/polymake/next/SparseMatrix.h
/usr/include/polymake/next/SparseVector.h
/usr/include/polymake/next/TropicalNumber.h
/usr/include/polymake/next/Vector.h
/usr/include/polymake/next/color.h
/usr/include/polymake/next/hash_map
/usr/include/polymake/next/hash_set
/usr/include/polymake/next/integer_linalg.h
/usr/include/polymake/next/linalg.h
/usr/include/polymake/next/list
/usr/include/polymake/next/numerical_functions.h