Irredundant Triangular Decomposition

Triangular decomposition is a classic, widely used and well-developed way to represent algebraic varieties with many applications. In particular, there exist sharp degree bounds for a single triangular set in terms of intrinsic data of the variety it represents, and powerful randomized algorithms for computing triangular decompositions using Hensel lifting in the zero-dimensional case and for irreducible varieties. However, in the general case, most of the algorithms computing triangular decompositions produce embedded components, which makes it impossible to directly apply the intrinsic degree bounds. This, in turn, is an obstacle for efficiently applying Hensel lifting due to the higher degrees of the output polynomials and the lower probability of success. In this paper, we give an algorithm to compute an irredundant triangular decomposition of an arbitrary algebraic set $W$ defined by a set of polynomials in C[x_1, x_2, ..., x_n]. Using this irredundant triangular decomposition, we were able to give intrinsic degree bounds for the polynomials appearing in the triangular sets and apply Hensel lifting techniques. Our decomposition algorithm is randomized, and we analyze the probability of success

Spanning forests and the q-state Potts model in the limit q \to 0

We study the q-state Potts model with nearest-neighbor coupling v=e^{\beta J}-1 in the limit q,v \to 0 with the ratio w = v/q held fixed. Combinatorially, this limit gives rise to the generating polynomial of spanning forests; physically, it provides information about the Potts-model phase diagram in the neighborhood of (q,v) = (0,0). We have studied this model on the square and triangular lattices, using a transfer-matrix approach at both real and complex values of w. For both lattices, we have computed the symbolic transfer matrices for cylindrical strips of widths 2 \le L \le 10, as well as the limiting curves of partition-function zeros in the complex w-plane. For real w, we find two distinct phases separated by a transition point w=w_0, where w_0 = -1/4 (resp. w_0 = -0.1753 \pm 0.0002) for the square (resp. triangular) lattice. For w > w_0 we find a non-critical disordered phase, while for w < w_0 our results are compatible with a massless Berker-Kadanoff phase with conformal charge c = -2 and leading thermal scaling dimension x_{T,1} = 2 (marginal operator). At w = w_0 we find a "first-order critical point": the first derivative of the free energy is discontinuous at w_0, while the correlation length diverges as w \downarrow w_0 (and is infinite at w = w_0). The critical behavior at w = w_0 seems to be the same for both lattices and it differs from that of the Berker-Kadanoff phase: our results suggest that the conformal charge is c = -1, the leading thermal scaling dimension is x_{T,1} = 0, and the critical exponents are \nu = 1/d = 1/2 and \alpha = 1.Comment: 131 pages (LaTeX2e). Includes tex file, three sty files, and 65 Postscript figures. Also included are Mathematica files forests_sq_2-9P.m and forests_tri_2-9P.m. Final journal versio

On the complexity of computing with zero-dimensional triangular sets

We study the complexity of some fundamental operations for triangular sets in dimension zero. Using Las-Vegas algorithms, we prove that one can perform such operations as change of order, equiprojectable decomposition, or quasi-inverse computation with a cost that is essentially that of modular composition. Over an abstract field, this leads to a subquadratic cost (with respect to the degree of the underlying algebraic set). Over a finite field, in a boolean RAM model, we obtain a quasi-linear running time using Kedlaya and Umans' algorithm for modular composition. Conversely, we also show how to reduce the problem of modular composition to change of order for triangular sets, so that all these problems are essentially equivalent. Our algorithms are implemented in Maple; we present some experimental results

On the scaling limits of planar percolation

We prove Tsirelson's conjecture that any scaling limit of the critical planar percolation is a black noise. Our theorems apply to a number of percolation models, including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid. We also suggest a natural construction for the scaling limit of planar percolation, and more generally of any discrete planar model describing connectivity properties.Comment: With an Appendix by Christophe Garban. Published in at http://dx.doi.org/10.1214/11-AOP659 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Limit Theorems for Estimating the Parameters of Differentiated Product Demand Systems

We provide an asymptotic distribution theory for a class of Generalized Method of Moments estimators that arise in the study of differentiated product markets when the number of observations is associated with the number of products within a given market. We allow for three sources of error: the sampling error in estimating market shares, the simulation error in approximating the shares predicted by the model, and the underlying model error. The limiting distribution of the parameter estimator is normal provided the size of the consumer sample and the number of simulation draws grow at a large enough rate relative to the number of products. We specialise our distribution theory to the Berry, Levinsohn, and Pakes (1995) random coefficient logit model and a pure characteristic model. The required rates differ for these two frequently used demand models. A small Monte Carlo study shows that the difference in asymptotic properties of the two models are reflected in the models' small sample properties. These differences impact directly on the computational burden of the two models.Choice models, Method of moments, Random coefficients, Product differentiation