A refined error analysis for fixed-degree polynomial optimization over the simplex

We consider the problem of minimizing a fixed-degree polynomial over the standard simplex. This problem is well known to be NP-hard, since it contains the maximum stable set problem in combinatorial optimization as a special case. In this paper, we revisit a known upper bound obtained by taking the minimum value on a regular grid, and a known lower bound based on P\'olya's representation theorem. More precisely, we consider the difference between these two bounds and we provide upper bounds for this difference in terms of the range of function values. Our results refine the known upper bounds in the quadratic and cubic cases, and they asymptotically refine the known upper bound in the general case.Comment: 13 page

Doubled Conformal Compactification

We use Weyl transformations between the Minkowski spacetime and dS/AdS spacetime to show that one cannot well define the electrodynamics globally on the ordinary conformal compactification of the Minkowski spacetime (or dS/AdS spacetime), where the electromagnetic field has a sign factor (and thus is discountinuous) at the light cone. This problem is intuitively and clearly shown by the Penrose diagrams, from which one may find the remedy without too much difficulty. We use the Minkowski and dS spacetimes together to cover the compactified space, which in fact leads to the doubled conformal compactification. On this doubled conformal compactification, we obtain the globally well-defined electrodynamics.Comment: 14 pages, 4 figure