A Dichotomy Theorem for Homomorphism Polynomials

In the present paper we show a dichotomy theorem for the complexity of polynomial evaluation. We associate to each graph H a polynomial that encodes all graphs of a fixed size homomorphic to H. We show that this family is computable by arithmetic circuits in constant depth if H has a loop or no edge and that it is hard otherwise (i.e., complete for VNP, the arithmetic class related to #P). We also demonstrate the hardness over the rational field of cut eliminator, a polynomial defined by B\"urgisser which is known to be neither VP nor VNP-complete in the field of two elements, if VP is not equal to VNP (VP is the class of polynomials computable by arithmetic circuit of polynomial size)

Depth-4 Lower Bounds, Determinantal Complexity : A Unified Approach

Tavenas has recently proved that any n^{O(1)}-variate and degree n polynomial in VP can be computed by a depth-4 circuit of size 2^{O(\sqrt{n}\log n)}. So to prove VP not equal to VNP, it is sufficient to show that an explicit polynomial in VNP of degree n requires 2^{\omega(\sqrt{n}\log n)} size depth-4 circuits. Soon after Tavenas's result, for two different explicit polynomials, depth-4 circuit size lower bounds of 2^{\Omega(\sqrt{n}\log n)} have been proved Kayal et al. and Fournier et al. In particular, using combinatorial design Kayal et al.\ construct an explicit polynomial in VNP that requires depth-4 circuits of size 2^{\Omega(\sqrt{n}\log n)} and Fournier et al.\ show that iterated matrix multiplication polynomial (which is in VP) also requires 2^{\Omega(\sqrt{n}\log n)} size depth-4 circuits. In this paper, we identify a simple combinatorial property such that any polynomial f that satisfies the property would achieve similar circuit size lower bound for depth-4 circuits. In particular, it does not matter whether f is in VP or in VNP. As a result, we get a very simple unified lower bound analysis for the above mentioned polynomials. Another goal of this paper is to compare between our current knowledge of depth-4 circuit size lower bounds and determinantal complexity lower bounds. We prove the that the determinantal complexity of iterated matrix multiplication polynomial is \Omega(dn) where d is the number of matrices and n is the dimension of the matrices. So for d=n, we get that the iterated matrix multiplication polynomial achieves the current best known lower bounds in both fronts: depth-4 circuit size and determinantal complexity. To the best of our knowledge, a \Theta(n) bound for the determinantal complexity for the iterated matrix multiplication polynomial was known only for constant d>1 by Jansen.Comment: Extension of the previous uploa

Thomas-Ehrman shifts in nuclei around ^{16}O and role of residual nuclear interaction

The asymmetry in the energy spectra between mirror nuclei (the Thomas-Ehrman shifts) around $^{16}$O is investigated from a phenomenological viewpoint. The recent data on proton-rich nuclei indicates that the residual nuclear interaction is reduced for the loosely bound s-orbit by as much as 30%, which originates in the broad radial distribution of the proton single-particle wave function.Comment: to appear in Phys. Lett. B, with 3 eps figure