pinecone/core-2020-05-10-deduplication huggingface dataset:
Reduced Gröbner Bases Under Composition

AbstractIn this paper we contribute with one main result to the interesting problem initiated by Hong (1998, J. Symb. Comput.25, 643–663) on the behaviour of Gröbner bases under composition of polynomials. Polynomial composition is the operation of replacing the variables of a polynomial with other polynomials. The main question of this paper is:When does composition commute with reduced Gröbner bases computation under the same term ordering?We give a complete answer for this question: let Θ be a polynomial map, then for every reduced Gröbner basisG,G○Θ is a reduced Gröbner basis if and only if the composition by Θ is compatible with the term ordering and Θ is a list of permuted univariate and monic polynomials. Besides, we also include other minor results concerned with this problem; in particular, we provide a sufficient condition to determine when composition commutes with reduced Gröbner bases computation (possibly) under different term ordering
Classification of Hermitian Forms with the Neighbour Method

AbstractThe neighbour method of Kneser can be adapted to the hermitian case. Generalizing results of Hoffmann (1991), we show that it can be used to classify any genus in a hermitian space of dimension ≥2 by neighbour steps at suitable primes. The method was implemented for positive definite hermitian lattices (not necessarily free) over Q(d1/2). A table of class numbers of unimodular genera and the largest minima attained in those genera is given. We also describe a generalization of the LLL-algorithm to lattices in positive hermitian spaces over number fields
Efficient Algorithms for Computing the Jacobi Symbol

AbstractWe present two new algorithms for computing the Jacobi Symbol: the right-shift and left-shiftk-ary algorithms. For inputs of at mostnbits in length, both algorithms takeO(n2/logn) time andO(n) space. This is asymptotically faster than the traditional algorithm, which is based in Euclid's algorithm for computing greatest common divisors. In practice, we found our new algorithms to be about twice as fast for inputs of 100 to 1000 decimal digits in length. We also present parallel versions of both algorithms for the CRCW PRAM. One version takesOϵ(n/ loglogn) time usingO(n1+ϵ) processors, giving the first sublinear parallel algorithms for this problem. The other version takes polylog time using a subexponential number of processors
Factoring Modular Polynomials

AbstractThis paper gives an algorithm to factor a polynomialf(in one variable) over rings like Z/rZ forr∈Z or F[y]/rF[y] forr∈F [y]. The Chinese Remainder Theorem reduces our problem to the case whereris a prime power. Then factorization is not unique, but ifrdoes not divide the discriminant off, our (probabilistic) algorithm produces a description of all (possibly exponentially many) factorizations into irreducible factors in polynomial time. Ifrdivides the discriminant, we only know how to factor by exhaustive search, in exponential time
A Normal Form for Function Rings of Piecewise Functions

AbstractComputer algebra systems often have to deal with piecewise continuous functions. These are, for example, the absolute value function, signum, piecewise defined functions but also functions that are the supremum or infimum of two functions. We present a new algebraic approach to these types of problems. This paper presents a normal form for a function ring containing piecewise polynomial functions of a real variable. We give a complete rule system to compute the normal form of an expression. The main result is that this normal form can be used to decide extensional equality of two piecewise functions. Also we define supremum and infimum for piecewise functions; in fact, we show that the function ring forms a lattice. Additionally, a method to solve equalities and inequalities in this function ring is presented. Finally, we give a “user interface” to the algebraic representation of the piecewise functions
Infinite Convergent String-rewriting Systems and Cross-sections for Finitely Presented Monoids

AbstractA finitely presented monoid has a decidable word problem if and only if it can be presented by some left-recursive convergent string-rewriting system if and only if it has a recursive cross-section. However, regular cross-sections or even context-free cross-sections do not suffice. This is shown by presenting examples of finitely presented monoids with decidable word problems that do not admit regular cross-sections, and that, hence, cannot be presented by left-regular convergent string-rewriting systems. Also examples of finitely presented monoids with decidable word problems are presented that do not even admit context-free cross-sections. On the other hand, it is shown that each finitely presented monoid with a decidable word problem has a finite presentation that admits a cross-section which is a Church–Rosser language. Finally we address the notion of left-regular convergent string-rewriting systems that are tractable
On Approximate GCDs of Univariate Polynomials

AbstractIn this paper, we consider computations involving polynomials with inexact coefficients, i.e. with bounded coefficient errors. The presence of input errors changes the nature of questions traditionally asked in computer algebra. For instance, given two polynomials, instead of trying to compute their greatest common divisor, one might now try to compute a pair of polynomials with a non-trivial common divisor close to the input polynomials. We consider the problem of finding approximate common divisors in the context of inexactly specified polynomials. We develop efficient algorithms for the so-called nearest common divisor problem and several of its variants
When are Two Numerical Polynomials Relatively Prime?

AbstractLet a and b be two polynomials having numerical coefficients. We consider the question: When are a and b relatively prime? Since the coefficients of a and b are approximant, the question is the same as: When are two polynomials relatively prime, even after small perturbations of the coefficients?In this paper we provide a numeric parameter for determining whether two polynomials are prime, even under small perturbations of the coefficients. Our methods rely on an inversion formula for Sylvester matrices to establish an effective criterion for relative primeness. The inversion formula can also be used to approximate the condition number of a Sylvester matrix
A Fast and Numerically Stable Euclidean-like Algorithm for Detecting Relatively Prime Numerical Polynomials

AbstractIn this paper we provide a fast, numerically stable algorithm to determine when two given polynomialsa and b are relatively prime and remain relatively prime even after small perturbations of their coefficients. Such a problem is important in many applications where input data are only available up to a certain precision.Our method—an extension of the Cabay–Meleshko algorithm for Padé approximation—is typically an order of magnitude faster than previously known stable methods. As such it may be used as an inexpensive test which may be applied before attempting to compute a “numerical GCD”, in general a much more difficult task. We prove that the algorithm is numerically stable and give experiments verifying the numerical behaviour. Finally, we discuss possible extensions of our approach that can be applied to the problem of actually computing a numerical GCD
Computing the Isolated Roots by Matrix Methods

AbstractTwo main approaches are used, nowadays, to compute the roots of a zero-dimensional polynomial system. The first one involves Gröbner basis computation, and applies to any zero-dimensional system. But, it is performed withexact arithmetic and, usually, large numbers appear during the computation. The other approach is based on resultant formulations and can be performed with floating point arithmetic. However, it applies only to generic situations, leading to singular problems in several systems coming from robotics and computational vision, for instance.In this paper, reinvestigating the resultant approach from the linear algebra point of view, we handle the problem of genericity and present a new algorithm for computing the isolated roots of an algebraic variety, not necessarily of dimension zero. We analyse two types of resultant formulations, transform them into eigenvector problems, and describe special linear algebra operations on the matrix pencils in order to reduce the root computation to a non-singular eigenvector problem. This new algorithm, based on pencil decompositions, has a good complexity even in the non-generic situations and can be executed with floating point arithmetic
A Symbolic-Numerical Method for Finding a Real Solution of an Arbitrary System of Nonlinear Algebraic Equations

AbstractIn this paper, the author presents a new method for iteratively finding a real solution of an arbitrary system of nonlinear algebraic equations, where the system can be overdetermined or underdetermined and its Jacobian matrix can be of any (positive) rank. When the number of equations is equal to the number of variables and the Jacobian matrix of the system is nonsingular, the method is similar to the well-known Newton's method.The method is a hybrid symbolic-numerical method, in that we utilize some extended procedures from classical computer algebra together with ideas and algorithmic techniques from numerical computation, namely Newton's method and pseudoinverse matrices. First the symbolic techniques are used to transform an arbitrary system of algebraic equations into a set of regular systems. By regular system we mean a system whose Jacobian matrix is of full row rank. Newton-like numerical techniques are then used to find a real solution for each regular system obtained from the symbolic part of the method.The method has a wide range of applicability. It is especially useful for applications in which we need to find some particular solutions from a nonzero-dimensional manifold of real solutions of a system of equations, i.e. the system has infinitely many solutions.We find some mild conditions for the asymptotic convergence of the numerical part of our method. We prove that the asymptotic convergence of the new method is still quadratic while the robustness of the numerical part can be enhanced by techniques like damping as in the regular case. The method has been implemented in Maple andMathematica . Several examples are presented to show that the method works nicely
Remarks on Automatic Algorithm Stabilization

AbstractThis note is about a methodology utilizing inexact computation in conjunction with exact computation where the exact input is known and exact output is desired. The inexact computation is used to help avert the growth of intermediate expressions. This growth frequently makes using exact computation infeasible. We mention several existing applications and also mention where the methodology is not useful. We propose new directions where one can make effective use of the stabilization methodology
Numerical Schubert Calculus

AbstractWe develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface Schubert conditions we give two algorithms based on extrinsic deformations of the Grassmannian: one is derived from a Gröbner basis for the Plücker ideal of the Grassmannian and the other from a SAGBI basis for its projective coordinate ring. The more general case of special Schubert conditions is solved by delicate intrinsic deformations, called Pieri homotopies, which first arose in the study of enumerative geometry over the real numbers. Computational results are presented and applications to control theory are discussed
Riemann Surfaces, Plane Algebraic Curves and Their Period Matrices

AbstractThe aim of this paper is to present theoretical basis for computing a representation of a compact Riemann surface as an algebraic plane curve and to compute a numerical approximation for its period matrix. We will describe a program C ars (Semmler et al., 1996) that can be used to define Riemann surfaces for computations. C ars allows one also to perform the Fenchel–Nielsen twist and other deformations on Riemann surfaces.Almost all theoretical results presented here are well known in classical complex analysis and algebraic geometry. The contribution of the present paper is the design of an algorithm which is based on the classical results and computes first an approximation of a polynomial representing a given compact Riemann surface as a plane algebraic curve and further computes an approximation for a period matrix of this curve. This algorithm thus solves an important problem in the general case. This problem was first solved, in the case of symmetric Riemann surfaces, in Seppälä (1994)
Computing Automatic Coset Systems and Subgroup Presentations

AbstractThe concept of an automatic group can be generalized to a group that is automatic with respect to a specified subgroup. This means that there is a finite state automaton that recognizes a unique word in each coset of the subgroup, and others that essentially recognize the permutation action on these cosets induced by multiplying by a group generator. These automata make it possible to enumerate coset representatives as words in the generators, and to solve the generalized word problem for the subgroup efficiently. Algorithms to construct these automata have been described previously by Redfern. Here we describe improved versions, together with implementation details and some examples of successful calculations. A related algorithm to compute a finite presentation of the subgroup is also described
A Unified Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems

AbstractWe prove that in all but one case the normal form of a real or complex Hamiltonian matrix which is irreducible and appropriately normalized can be computed by Lie series methods in formally the same manner as one computes the normal form of a nonlinear Hamiltonian function. Calculations are emphasized; the methods are illustrated with detailed examples, and for the sake of completeness the exceptional case is also reviewed and illustrated. Alternate methods are also discussed, along with detailed examples
Monomial Orderings, Rewriting Systems, and Gröbner Bases for the Commutator Ideal of a Free Algebra

AbstractIn this paper we consider a free associative algebra on three generators over an arbitrary fieldK. Given a term ordering on the commutative polynomial ring on three variables overK, we construct uncountably many liftings of this term ordering to a monomial ordering on the free associative algebra. These monomial orderings are total well orderings on the set of monomials, resulting in a set of normal forms. Then we show that the commutator ideal has an infinite reduced Gröbner basis with respect to these monomial orderings, and all initial ideals are distinct. Hence, the commutator ideal has at least uncountably many distinct reduced Gröbner bases. A Gröbner basis of the commutator ideal corresponds to a complete rewriting system for the free commutative monoid on three generators; our result also shows that this monoid has at least uncountably many distinct minimal complete rewriting systems.The monomial orderings we use are not compatible with multiplication, but are sufficient to solve the ideal membership problem for a specific ideal, in this case the commutator ideal. We propose that it is fruitful to consider such, more general, monomial orderings in non-commutative Gröbner basis theory
Basic Algorithms for Rational Function Fields

AbstractBy means of Gröbner basis techniques algorithms for solving various problems concerning subfields K(g):=K(g1, …,gm) of a rational function field K(x):=K(x1, …,xn) are derived: computing canonical generating sets, deciding field membership, computing the degree and separability degree resp. the transcendence degree and a transcendence basis of K(x)/K(g), deciding whetherf∈K(x) is algebraic or transcendental over K(g), computing minimal polynomials, and deciding whether K(g) contains elements of a “particular structure”, e.g. monic univariate polynomials of fixed degree. The essential idea is to reduce these problems to questions concerning an ideal of a polynomial ring; connections between minimal primary decompositions over K(x) of this ideal and intermediate fields of K(g) and K(x) are given. In the last section some practical considerations concerning the use of the algorithms are discussed
An Algorithm to Calculate Optimal Homogeneous Systems of Parameters

AbstractWhen a homogeneous system of parametersf1,...,fnis chosen for a graded algebraA, it is important for subsequent computations that the degrees, deg(fi), are as small as possible. More precisely, one would like the product or the sum of the degrees to be minimal, depending on the application.This article investigates which degree vectors can occur as the degrees of a homogeneous system of parameters. From this, an algorithm is derived which constructs an optimal homogeneous system of parameters. Here the notion of what is considered asoptimalis part of the input. An important application is the case whereAis the invariant ring of a finite linear group. There is an implementation of the algorithm in Magma which applies to this case
Lazy Linear Combinatorial Unification

AbstractIn this paper we define a two-rule reduction system for the lazy computation of unification on first-order terms. The computation consists of a sequence of reductions which is almost linear with the size of the input. Every reduction transforms the problem into an equivalent one which is a better approximation of the solution. The approach is based on a combinatorial system which provides abstractions for terms and term unification, and does not resort to any specific reduction control strategy. The abstraction on both data and control makes our system suitable in any applicative context where unification is required
Computer Algebra Algorithms and Routines for the Computation of Conservation Laws and Fixing of Gauge in Differential Expressions

AbstractIn this paper we present three different approaches for the determination of conservation laws. For three correspondingREDUCEcomputer algebra programsCONLAW1/2/3the necessary subroutines are described. All three programs use subroutines which remove redundant functions and constants in the general solution of the conservation law conditions. The corresponding algorithm is explained. Such a program is not only applicable in the context of computing conservation laws but whenever redundancy in differential expressions is to be removed or gauge freedom to be fixed
A Framework for Operational Equational Specifications with Pre-defined Structures

AbstractWe present a general framework for studying equational specifications with pre-defined structures. The axioms of the specifications are to define new structures in addition to the given ones. In particular, they may define a new operator only partially over some given domain. Our approach allows one to assign easily semantics to such specifications in a denotational and operational fashion. In order to enable functional-style computations, we introduce a semantically enriched notion of term rewriting. This rewrite relation also allows us to infer the consistency of the specification. For the latter purpose one has to show confluence modulo the given structures. We outline how to obtain criteria easily for confluence and termination of the rewrite relation of discourse by generalizing results of the classical syntactic rewrite theory
A Transducer Approach to Coxeter Groups

AbstractThis paper represents an efficient way of implementing a (finite) Coxeter groupWon a computer, through the use of a tower of special subgroups ofWand the corresponding left cosets. We show how some results of Deodhar's may be interpreted in the language of finite state machines to produce a cascade of small transducers which handle the main “word processing” problems in the group. An algorithm is described for an efficient explicit construction of the relevant transducer tables starting from the Coxeter matrix ofW. We also show (again interpreting a result of Deodhar's) that the Bruhat ordering onWpresents a nice recursive structure in this implementation, leading to an effective enumeration of arbitrary Bruhat intervals
Construction of Finite Groups

AbstractWe1999 Academic Pressintroduce three practical algorithms to construct certain finite groups up to isomorphism. The first one can be used to construct all soluble groups of a given order. This method can be restricted to compute the soluble groups with certain properties such as nilpotent, non-nilpotent or supersoluble groups. The second algorithm can be used to determine the groups of orderCopyright pn· qwith a normal Sylow subgroup for distinct primespandq. The third method is a general method to construct finite groups which we use to compute insoluble groups
Matrices in Elimination Theory

AbstractThe last decade has witnessed the rebirth of resultant methods as a powerful computational tool for variable elimination and polynomial system solving. In particular, the advent of sparse elimination theory and toric varieties has provided ways to exploit the structure of polynomials encountered in a number of scientific and engineering applications. On the other hand, the Bezoutian reveals itself as an important tool in many areas connected to elimination theory and has its own merits, leading to new developments in effective algebraic geometry. This survey unifies the existing work on resultants, with emphasis on constructing matrices that generalize the classic matrices named after Sylvester, Bézout and Macaulay. The properties of the different matrix formulations are presented, including some complexity issues, with emphasis on variable elimination theory. We compare toric resultant matrices to Macaulay’s matrix and further conjecture the generalization of Macaulay’s exact rational expression for the resultant polynomial to the toric case. A new theorem proves that the maximal minor of a Bézout matrix is a non-trivial multiple of the resultant. We discuss applications to constructing monomial bases of quotient rings and multiplication maps, as well as to system solving by linear algebra operations. Lastly, degeneracy issues, a major preoccupation in practice, are examined. Throughout the presentation, examples are used for illustration and open questions are stated in order to point the way to further research
Simultaneous Elimination by using Several Tools from Real Algebraic Geometry

AbstractThe aim of this paper is to introduce two new elimination procedures for algebraic systems of equations. The first one eliminates one variable from a finite set of polynomials with complex or real coefficients and it is based on a parametric version of Barnett’s Method for computing the greatest common divisor of a finite family of univariate polynomials. The second one, based on Hermite’s Method, deals with the global elimination of a block of variables from a finite set of multivariate polynomials with a particular structure (containing a Pham system). A common feature of both procedures is that the final step relies on a specific property of a real-valued inner product on vector spaces over the coefficient field: Gram’s Criterion