An Attempt to Enhance Buchberger's Algorithm by Using Remainder Sequences and GCDs (II) (Computer Algebra - Theory and its Applications)

Let F = {F, , ..., Fm+1} ⊂ ℚ[x, u] be a given system, where m+l 2: 3, (x) = (x, , ..., xm) and (u) = (u, , ...，叫）， with ∀xi >-- ∀uj. Let GB(F) = {G₁, G₂, ・・・}, with G₁ --< G₂ --< ・・・, be the reduced Grabner basis of F w.r.t. the lexicographic order. In a previous paper [10], one of the authors proposed a method of enhancing Buchberger's algorithm for computing GB(F). His idea_is to compute a set g':= {G1 , G2, ... } ⊂ ℚ[x, u], such that each Gi is either O or as mall multiple of Gi, and apply Buchberger's algorithm to F ∨ g'. He proposed a scheme of computing G₁, G₂, ... by the PRSs (polynomial remainder sequences) and the GCDs in "G₁ ⇒ G₂ ⇒ ・・・" order, without computing Spolynomials. The scheme is supported by two new useful theorems and one proposition to remove the extraneous factor. In fact, for a simple but never toy example, his scheme has computed G₁ successfully (G₁ became G₁ by the proposition mentioned above). However, an unexpected difficulty occurred in computing G₂; it contained a pretty large extraneous factor which was not removed by the proposition. In this paper, we find a surprising phenomenon with which we can remove the above mentioned extraneous factor in G₂ and obtain G₂. As for G₃ and G₄, we obtain very good "body doubles" of them, by eliminating variables in leading coefficients of intermediate remainders of the PRSs computed for G₁. For systems of many sub-variables, n ≥ 3, our method introduces an extra factor in ℚ[u3, ..，un]， into the "LCto W" polynomial; see the text for the LCtoW polynomial. Furthermore, we present several techniques to enhance the computation

The Complexity of Algebraic Algorithms for LWE

Arora & Ge introduced a noise-free polynomial system to compute the secret of a Learning With Errors (LWE) instance via linearization. Albrecht et al. later utilized the Arora-Ge polynomial model to study the complexity of Gr\"obner basis computations on LWE polynomial systems under the assumption of semi-regularity. In this paper we revisit the Arora-Ge polynomial and prove that it satisfies a genericity condition recently introduced by Caminata & Gorla, called being in generic coordinates. For polynomial systems in generic coordinates one can always estimate the complexity of DRL Gr\"obner basis computations in terms of the Castelnuovo-Mumford regularity and henceforth also via the Macaulay bound. Moreover, we generalize the Gr\"obner basis algorithm of Semaev & Tenti to arbitrary polynomial systems with a finite degree of regularity. In particular, existence of this algorithm yields another approach to estimate the complexity of DRL Gr\"obner basis computations in terms of the degree of regularity. In practice, the degree of regularity of LWE polynomial systems is not known, though one can always estimate the lowest achievable degree of regularity. Consequently, from a designer's worst case perspective this approach yields sub-exponential complexity estimates for general, binary secret and binary error LWE. In recent works by Dachman-Soled et al. the hardness of LWE in the presence of side information was analyzed. Utilizing their framework we discuss how hints can be incorporated into LWE polynomial systems and how they affect the complexity of Gr\"obner basis computations

The computational complexity of the Chow form

We present a bounded probability algorithm for the computation of the Chow forms of the equidimensional components of an algebraic variety. Its complexity is polynomial in the length and in the geometric degree of the input equation system defining the variety. In particular, it provides an alternative algorithm for the equidimensional decomposition of a variety. As an application we obtain an algorithm for the computation of a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents. As a further application, we derive an algorithm for the computation of the (unique) solution of a generic over-determined equation system.Comment: 60 pages, Latex2

Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)

Oxford, UK, 26 August 200

Fast and Stable Polynomial Equation Solving and Its Application to Computer Vision

This paper presents several new results on techniques for solving systems of polynomial equations in computer vision. Gröbner basis techniques for equation solving have been applied successfully to several geometric computer vision problems. However, in many cases these methods are plagued by numerical problems. In this paper we derive a generalization of the Gröbner basis method for polynomial equation solving, which improves overall numerical stability. We show how the action matrix can be computed in the general setting of an arbitrary linear basis for ℂ[x]/I. In particular, two improvements on the stability of the computations are made by studying how the linear basis for ℂ[x]/I should be selected. The first of these strategies utilizes QR factorization with column pivoting and the second is based on singular value decomposition (SVD). Moreover, it is shown how to improve stability further by an adaptive scheme for truncation of the Gröbner basis. These new techniques are studied on some of the latest reported uses of Gröbner basis methods in computer vision and we demonstrate dramatically improved numerical stability making it possible to solve a larger class of problems than previously possible

A Differential Algebra Introduction For Tropical Differential Geometry

Doctora