Implicitization of rational surfaces using toric varieties

A parameterized surface can be represented as a projection from a certain toric surface. This generalizes the classical homogeneous and bihomogeneous parameterizations. We extend to the toric case two methods for computing the implicit equation of such a rational parameterized surface. The first approach uses resultant matrices and gives an exact determinantal formula for the implicit equation if the parameterization has no base points. In the case the base points are isolated local complete intersections, we show that the implicit equation can still be recovered by computing any non-zero maximal minor of this matrix. The second method is the toric extension of the method of moving surfaces, and involves finding linear and quadratic relations (syzygies) among the input polynomials. When there are no base points, we show that these can be put together into a square matrix whose determinant is the implicit equation. Its extension to the case where there are base points is also explored.Comment: 28 pages, 1 figure. Numerous major revisions. New proof of method of moving surfaces. Paper accepted and to appear in Journal of Algebr

The implicit equation of a multigraded hypersurface

In this article we analyze the implicitization problem of the image of a rational map $\phi: X --> P^n$, with $T$ a toric variety of dimension $n-1$ defined by its Cox ring $R$. Let $I:=(f_0,...,f_n)$ be $n+1$ homogeneous elements of $R$. We blow-up the base locus of $\phi$, $V(I)$, and we approximate the Rees algebra $Rees_R(I)$ of this blow-up by the symmetric algebra $Sym_R(I)$. We provide under suitable assumptions, resolutions $\Z.$ for $Sym_R(I)$ graded by the torus-invariant divisor group of $X$, $Cl(X)$, such that the determinant of a graded strand, $\det((\Z.)_\mu)$, gives a multiple of the implicit equation, for suitable $\mu\in Cl(X)$. Indeed, we compute a region in $Cl(X)$ which depends on the regularity of $Sym_R(I)$ where to choose $\mu$. We also give a geometrical interpretation of the possible other factors appearing in $\det((\Z.)_\mu)$. A very detailed description is given when $X$ is a multiprojective space.Comment: 19 pages, 2 figures. To appear in Journal of Algebr

Vectorization on the star computer of several numerical methods for a fluid flow problem

A reexamination of some numerical methods is considered in light of the new class of computers which use vector streaming to achieve high computation rates. A study has been made of the effect on the relative efficiency of several numerical methods applied to a particular fluid flow problem when they are implemented on a vector computer. The method of Brailovskaya, the alternating direction implicit method, a fully implicit method, and a new method called partial implicitization have been applied to the problem of determining the steady state solution of the two-dimensional flow of a viscous imcompressible fluid in a square cavity driven by a sliding wall. Results are obtained for three mesh sizes and a comparison is made of the methods for serial computation

An Output-sensitive Algorithm for Computing Projections of Resultant Polytopes

We develop an incremental algorithm to compute the Newton polytope of the resultant, aka resultant polytope, or its projection along a given direction. The resultant is fundamental in algebraic elimination and in implicitization of parametric hypersurfaces. Our algorithm exactly computes vertex- and halfspace-representations of the desired polytope using an oracle producing resultant vertices in a given direction. It is output-sensitive as it uses one oracle call per vertex. We overcome the bottleneck of determinantal predicates by hashing, thus accelerating execution from $18$ to $100$ times. We implement our algorithm using the experimental CGAL package {\tt triangulation}. A variant of the algorithm computes successively tighter inner and outer approximations: when these polytopes have, respectively, 90\% and 105\% of the true volume, runtime is reduced up to $25$ times. Our method computes instances of $5$-, $6$- or $7$-dimensional polytopes with $35$K, $23$K or $500$ vertices, resp., within $2$hr. Compared to tropical geometry software, ours is faster up to dimension $5$ or $6$, and competitive in higher dimensions

Causal inference via algebraic geometry: feasibility tests for functional causal structures with two binary observed variables

We provide a scheme for inferring causal relations from uncontrolled statistical data based on tools from computational algebraic geometry, in particular, the computation of Groebner bases. We focus on causal structures containing just two observed variables, each of which is binary. We consider the consequences of imposing different restrictions on the number and cardinality of latent variables and of assuming different functional dependences of the observed variables on the latent ones (in particular, the noise need not be additive). We provide an inductive scheme for classifying functional causal structures into distinct observational equivalence classes. For each observational equivalence class, we provide a procedure for deriving constraints on the joint distribution that are necessary and sufficient conditions for it to arise from a model in that class. We also demonstrate how this sort of approach provides a means of determining which causal parameters are identifiable and how to solve for these. Prospects for expanding the scope of our scheme, in particular to the problem of quantum causal inference, are also discussed.Comment: Accepted for publication in Journal of Causal Inference. Revised and updated in response to referee feedback. 16+5 pages, 26+2 figures. Comments welcom

Model testing for causal models

Finding cause-effect relationships is the central aim of many studies in the physical, behavioral, social and biological sciences. We consider two well-known mathematical causal models: Structural equation models and causal Bayesian networks. When we hypothesize a causal model, that model often imposes constraints on the statistics of the data collected. These constraints enable us to test or falsify the hypothesized causal model. The goal of our research is to develop efficient and reliable methods to test a causal model or distinguish between causal models using various types of constraints. For linear structural equation models, we investigate the problem of generating a small number of constraints in the form of zero partial correlations, providing an efficient way to test hypothesized models. We study linear structural equation models with correlated errors focusing on the graphical aspects of the models. We provide a set of local Markov properties and prove that they are equivalent to the global Markov property. For causal Bayesian networks, we study equality and inequality constraints imposed on data and investigate a way to use these constraints for model testing and selection. For equality constraints, we formulate an implicitization problem and show how we may reduce the complexity of the problem. We also study the algebraic structure of the equality constraints. For inequality constraints, we present a class of inequality constraints on both nonexperimental and interventional distributions