M Theory Lesson 348

This green book should be popular amongst physicists working on black hole entropy measures.

Recall that the set $A$ of points in the continuum of dimension $k - 1$ represented monomials in $k$ variables. A major theorem in the book shows that the secondary polytope of the (generalised) discriminant of $A$ may be expressed in terms of $A$ and its convex hull via triangulations, just as for the associahedra. In general, a vertex of this magic polytope, along with the simplex volumes of the triangulation, gives us a monomial of the discriminant, with its coefficient. The logarithm of the coefficient takes the form

$\sum V_{i} \textrm{log} V_{i}$

which is clearly an entropy in terms of the simplex volumes. For example, the triangles inside a (triangular) hexagon give us the coefficient for a $3d$ associahedron vertex. The authors point out that this entropy explains classical numbers such as the $27$ in the cubic discriminant of a polynomial. Later on in the book we find other instances of this non statistical entropy. In connection to discriminant hypersurfaces in tori, we see an entropy with the property that the sum $\sum V_{i}$ of volume terms equals zero. This suggests the possibility of negative volumes, or areas.

The final chapter covers general hyperdeterminants, including of course Cayley's classic example, now well known in studies of multiple qubit entanglement. It is encouraging to know that mathematicians have thought hard about computational tools associated to such general matrix objects.