On the irreducibility of multivariate subresultants

Let $P_1,...,P_n$ be generic homogeneous polynomials in $n$ variables of degrees $d_1,...,d_n$ respectively. We prove that if $\nu$ is an integer satisfying ${\sum_{i=1}^n d_i}-n+1-\min\{d_i\}<\nu,$ then all multivariate subresultants associated to the family $P_1,...,P_n$ in degree $\nu$ are irreducible. We show that the lower bound is sharp. As a byproduct, we get a formula for computing the residual resultant of $\binom{\rho-\nu +n-1}{n-1}$ smooth isolated points in \PP^{n-1}.Comment: Updated version, 4 pages, to appear in CRA

Feasible Computation in Symbolic and Numeric Integration

Two central concerns in scientific computing are the reliability and efficiency of algorithms. We introduce the term feasible computation to describe algorithms that are reliable and efficient given the contextual constraints imposed in practice. The main focus of this dissertation then, is to bring greater clarity to the forms of error introduced in computation and modeling, and in the limited context of symbolic and numeric integration, to contribute to integration algorithms that better account for error while providing results efficiently. Chapter 2 considers the problem of spurious discontinuities in the symbolic integration problem, proposing a new method to restore continuity based on a pair of unwinding numbers. Computable conditions for the unwinding numbers are specified, allowing the computation of a variety of continuous integrals. Chapter 3 introduces two structure-preserving algorithms for the symbolic-numeric integration of rational functions on exact input. A structured backward and forward error analysis for the algorithms shows that they are a posteriori backward and forward stable, with both forms of error exhibiting tolerance proportionality. Chapter 4 identifies the basic logical structure of feasible inference by presenting a logical model of stable approximate inference, illustrated by examples of modeling and numerical integration. In terms of this model it is seen that a necessary condition for the feasibility of methods of abstraction in modeling and complexity reduction in computational mathematics is the preservation of inferential structure, in a sense that is made precise. Chapter 5 identifies a robust pattern in mathematical sciences of transforming problems to make solutions feasible. It is showed that computational complexity reduction methods in computational science involve chains of such transformations. It is argued that the structured and approximate nature of such strategies indicates the need for a higher-order model of computation and a new definition of computational complexity

A Note on Subresultants and the Lazard/Rioboo/Trager Formula in Rational Function Integration

An ambiguity in a formula of Lazard, Rioboo and Trager, connecting subresultants and rational function integration, is indicated and examples of incorrect interpretations are given. 1 Introduction In [4] the authors present a formula connecting the logarithmic part of the integral of a rational function and certain subresultants. They also give an algorithm based on this formula. The same algorithm was implemented independently by Trager in SCRATCHPAD II (now AXIOM). In this paper it will be shown that this formula is ambiguous and that a wrong interpretation of this formula and the corresponding algorithm can lead to wrong results. In fact the formula has been wrongly interpreted in [3] and the wrong interpretation has been implemented in AXIOM 2.0. 2 The Formula Let K be a field of characteristic 0 and P (x); Q(x) 2 K[x] such that deg(P (x)) ! deg(Q(x)) and Q(x) square-free. Consider the differentiation 0 = d dx on K(x). Let S(y) 2 K[y] be the resultant of Q(x) and P (x) \Gamm..

Towards Comprehensive Parametric Code Generation Targeting Graphics Processing Units in Support of Scientific Computation

The most popular multithreaded languages based on the fork-join concurrency model (CIlkPlus, OpenMP) are currently being extended to support other forms of parallelism (vectorization, pipelining and single-instruction-multiple-data (SIMD)). In the SIMD case, the objective is to execute the corresponding code on a many-core device, like a GPGPU, for which the CUDA language is a natural choice. Since the programming concepts of CilkPlus and OpenMP are very different from those of CUDA, it is desirable to automatically generate optimized CUDA-like code from CilkPlus or OpenMP. In this thesis, we propose an accelerator model for annotated C/C++ code together with an implementation that allows the automatic generation of CUDA code. One of the key features of this CUDA code generator is that it supports the generation of CUDA kernel code where program parameters (like number of threads per block) and machine parameters (like shared memory size) are treated as unknown symbols. Hence, these parameters need not to be known at code-generation-time: machine parameters and program parameters can be respectively determined when the generated code is installed on the target machine. In addition, we show how these parametric CUDA programs can be optimized at compile-time in the form of a case discussion, where cases depend on the values of machine parameters (e.g. hardware resource limits) and program parameters (e.g. dimension sizes of thread-blocks). This generation of parametric CUDA kernels requires to deal with non-linear polynomial expressions during the dependence analysis and tiling phase. To achieve these algebraic calculations, we take advantage of techniques from computer algebra, in particular in the RegularChains library of Maple. Various illustrative examples are provided together with performance evaluation