Lower Bounds in Real Algebraic Geometry and Orientability of Real Toric Varieties

The real solutions to a system of sparse polynomial equations may be realized as a fiber of a projection map from a toric variety. When the toric variety is orientable, the degree of this map is a lower bound for the number of real solutions to the system of equations. We strengthen previous work by characterizing when the toric variety is orientable. This is based on work of Nakayama and Nishimura, who characterized the orientability of smooth real toric varieties.     

Exponential Gelfond-Khovanskii formula in dimension one

Gelfond and Khovanskii found a formula for the sum of the values of a Laurent polynomial over the zeros of a system of Laurent polynomials in . We expect that a similar formula holds in the case of exponential sums with real frequencies. Here we prove such a formula in dimension one.

     

Iodinated aluminum(III) corroles with long-lived triplet excited states

The first reported iodination of a corrole leads to selective functionalization of the four C-H bonds on one pole of the macrocycle. An aluminum(III) complex of the tetraiodinated corrole, which exhibits red fluorescence, possesses a long-lived triplet excited state.     

Lower bounds for real solutions to sparse polynomial systems

We show how to construct sparse polynomial systems that have non-trivial lower bounds on their numbers of real solutions. These are unmixed systems associated to certain polytopes. For the order polytope of a poset P this lower bound is the sign-imbalance of P and it holds if all maximal chains of P have length of the same parity. This theory also gives lower bounds in the real Schubert calculus through the sagbi degeneration of the Grassmannian to a toric variety, and thus recovers a result of Eremenko and Gabrielov.     

Minkowski Length of 3D Lattice Polytopes

We study the Minkowski length L(P) of a lattice polytope P, which is defined to be the largest number of non-trivial primitive segments whose Minkowski sum lies in P. The Minkowski length represents the largest possible number of factors in a factorization of polynomials with exponent vectors in P, and shows up in lower bounds for the minimum distance of toric codes. In this paper we give a polytime algorithm for computing L(P) where P is a 3D lattice polytope. We next study 3D lattice polytopes of Minkowski length 1. In particular, we show that if Q, a subpolytope of P, is the Minkowski sum of L=L(P) lattice polytopes Q _i , each of Minkowski length 1, then the total number of interior lattice points of the polytopes Q ₁,??,Q _L is at most 4. Both results extend previously known results for lattice polygons. Our methods differ substantially from those used in the two-dimensional case.