On the classification of quasihomogeneous functions

Abstract

We give a criterion for the existence of a non-degenerate quasihomogeneous polynomial in a configuration, i.e. in the space of polynomials with a fixed set of weights, and clarify the relation of this criterion to the necessary condition derived from the formula for the Poincaré polynomial. We further prove finiteness of the number of configurations for a given value of the singularity index. For the value 3 of this index, which is of particular interest in string theory, a constructive version of this proof implies an algorithm for the calculation of all non-degenerate configurations.

Periods for Calabi-Yau and Landau-Ginzburg vacua

“…Recently, a complete listing of non-degenerate Landau-Ginzburg potentials leading to N = 2 string vacua with c = 9 has been achieved [8][9][10]. 7,555 (about 3 4 ) of these models admit a standard geometrical interpretation in terms of hypersurfaces defined from the vanishing of a polynomial in a weighted IP 4 .…”

“…[11,9,10], we choose the reference polynomial P 0 (x i ) in Eq. (2.3) to be given by the sum of N monomials…”

Complete classification of reflexive polyhedra in four dimensions

“…In the present work we present the results of a complete classification in four dimensions, and as a side result we can state that the corresponding moduli space of Calabi-Yau threefolds is connected (this was almost, but not quite completely shown in [25]). We plan to make our complete results accessible at our web site [26].…”

Perspectives in High-Energy Physics

“…A popular suggestion is that the 6 surplus space dimensions are compactified on a Calabi-Yau manifold, but which one? 473 800 776 are known [74]! Recently we have embarked on a systematic study of Calabi-Yau (CY) spaces, constructed as zeroes of polynomials in weighted projective spaces, a technique which enables one to explore some of their internal properties and focus on these with desirable features [75].…”