Yuzhen Xie

Lifting techniques for triangular decompositions

We present lifting techniques for triangular decompositions of zero-dimensional varieties, that extend the range of the previous methods. We discuss complexity aspects, and report on a preliminary implementation. Our theoretical results are comforted by these experiments.

FFT-based dense polynomial arithmetic on multi-cores

We report efficient implementation techniques for FFT-based dense multivariate polynomial arithmetic over finite fields, targeting multi-cores. We have extended a preliminary study dedicated to polynomial multiplication and obtained a complete set of efficient parallel routines in Cilk++ for polynomial arithmetic such as normal form computation. Since bivariate multiplication applied to balanced data is a good kernel for these routines, we provide an in-depth study on the performance and the cut-off criteria of our different implementations for this operation. We also show that, not only optimized parallel multiplication can improve the performance of higher-level algorithms such as normal form computation but also this composition is necessary for parallel normal form computation to reach peak performance on a variety of problems that we have tested.

When does ( T ) equal sat( T )

Given a regular chain T, we aim at finding an efficient way for computing a system of generators of Sat(T), the saturated ideal of T. A natural idea is to test whether the equality {T}=Sat(T) holds, that is, whether T generates its saturated ideal. By generalizing the notion of primitivity from univariate polynomials to regular chains, we establish a necessary and sufficient condition, together with a Grobner basis free algorithm, for testing this equality. Our experimental results illustrate the efficiency of this approach in practice.

Efficient Computations of Irredundant Triangular Decompositions with the RegularChains Library

We present new functionalities that we have added to the RegularChains library in Maple to efficiently compute irredundant triangular decompositions. We report on the implementation of different strategies. Our experiments show that, for difficult input systems, the computing time for removing redundant components can be reduced to a small portion of the total time needed for solving these systems.

The RegularChains library in MAPLE

Performing calculations modulo a set of relations is a basictechnique in algebra. For instance, computing the inverse of aninteger modulo a prime integer or computing the inverse of thecomplex number 3 + 2<i>t</i> modulo the relation&ell;² + 1 = 0. Computing modulo a set<i>S</i> containing more than one relation requiresfrom <i>S</i> to have some mathematical structure. Forinstance, computing the inverse of <i>p</i> =<i>x</i> + <i>y</i> modulo<i>S</i> ={<i>x</i>² +<i>y</i> +1,<i>y</i>² +<i>x</i> + 1} is difficult unless one realizes thatthis question is equivalent to computing the inverse of<i>p</i> modulo <i>C</i> ={<i>x</i>⁴ +2<i>x</i>² +<i>x</i> + 2,<i>y</i> +<i>x</i>² + 1}. Indeed, fromthere one can simplify <i>p</i> using<i>y</i> =-<i>x</i>² - 1 leading to<i>q</i> =-<i>x</i>² +<i>x</i> - 1 and compute the inverse of<i>q</i> modulo<i>x</i>⁴ +2<i>x</i>² +<i>x</i> + 2 (using the extended Euclidean algorithm)leading to -1/2<i>x</i>³ -1/2<i>x</i>. One commonly used mathematical structurefor a set of algebraic relations is that of a<i>Gr&ouml;bner basis.</i> It is particularly wellsuited for deciding whether a quantity is null or not modulo a setof relations. For inverse computations, the notion of a<i>regular chain</i> is more adequate. For instance,computing the inverse of <i>p</i> =<i>x</i> + <i>y</i> modulo the set<i>C</i> ={<i>y</i>² -2<i>x</i> +1,<i>x</i>² -3<i>x</i> + 2}, which is both a Gr&ouml;bner basisand a regular chain, is easily answered in this latter point ofview. Indeed, it naturally leads to consider the GCD of<i>p</i> and<i>C<inf>y</inf></i> =<i>y</i>² -2<i>x</i> + 1 modulo the relation<i>C<inf>x</inf></i> =<i>x</i>² -3<i>x</i> + 2 = 0, which is [EQUATION] This shows that <i>p</i> has no inverse if<i>x</i> = 1 and has an inverse (which can be computedand which is -<i>y</i> + 2) if <i>x</i> =2.

Spiral-generated modular FFT algorithms

This paper presents an extension of the Spiral system to automatically generate and optimize FFT algorithms for the discrete Fourier transform over finite fields. The generated code is intended to support modular algorithms for multivariate polynomial computations in the modpn library used by Maple. The resulting code provides an order of magnitude speedup over the original implementations in the modpn library, and the Spiral system provides the ability to automatically tune the FFT code to different computing platforms.

BALANCED DENSE POLYNOMIAL MULTIPLICATION ON MULTI-CORES

In symbolic computation, polynomial multiplication is a fundamental operation akin to matrix multiplication in numerical computation. We present efficient implementation strategies for FFT-based dense polynomial multiplication targeting multi-cores. We show that balanced input data can maximize parallel speedup and minimize cache complexity for bivariate multiplication. However, unbalanced input data, which are common in symbolic computation, are challenging. We provide efficient techniques, that we call contraction and extension, to reduce multivariate (and univariate) multiplication to balanced bivariate multiplication. Our implementation in Cilk++ demonstrates good speedup on multi-cores.

Cache Complexity and Multicore Implementation for Univariate Real Root Isolation

We present parallel algorithms with optimal cache complexity for the kernel routine of many real root isolation algorithms, namely the Taylor shift by 1. We then report on multicore implementation for isolating the real roots of univariate polynomials with integer coefficients based on a classical algorithm due to Vincent, Collins and Akritas. For processing some well-known benchmark examples with sufficiently large size, our software tool reaches linear speedup on an 8-core machine. In addition, we show that our software is able to fully utilize the many cores and the memory space of a 32-core machine to tackle large problems that are out of reach for a desktop implementation.

The ConstructibleSetTools and ParametricSystemTools Modules of the RegularChains Library in Maple

We present two new modules of the regular chains library in Maple: constructible set tools which is the first distributed package dedicated to the maniputation of (parametric or not) constructible sets and parametric system tools which is the first implementation of comprehensive triangular decomposition. We illustrate the functionalities of these new modules by examples and describe our software design and implementation techniques. Since several existing packages have functionalities related to those of our new modules, we include an overview of the algorithms and software for manipulating constructible sets and solving parametric systems.

On the verification of polynomial system solvers

We discuss the verification of mathematical software solving polynomial systems symbolically by way of triangular decomposition. Standard verification techniques are highly resource consuming and apply only to polynomial systems which are easy to solve. We exhibit a new approach which manipulates constructible sets represented by regular systems. We provide comparative benchmarks of different verification procedures applied to four solvers on a large set of well-known polynomial systems. Our experimental results illustrate the high efficiency of our new approach. In particular, we are able to verify triangular decompositions of polynomial systems which are not easy to solve.