pinecone/core-2020-05-10-deduplication huggingface dataset:
Solving Degenerate Sparse Polynomial Systems Faster

AbstractConsider a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic. We present a fast method to find a point in every irreducible component of the zero set Z of F. Our techniques allow us to sharpen and lower prior complexity bounds for this problem by fully taking into account the monomial term structure. As a corollary of our development we also obtain new explicit formulae for the exact number of isolated roots of F and the intersection multiplicity of the positive-dimensional part of Z. Finally, we present a combinatorial construction of non-degenerate polynomial systems, with specified monomial term structure and maximally many isolated roots, which may be of independent interest
Finding All Isolated Zeros of Polynomial Systems inCnvia Stable Mixed Volumes

AbstractTo find all the isolated zeroes of a polynomial system P(x) inCn (as opposed to in (C*)n) via the polyhedral homotopy of Huber and Sturmfels (1997), one first finds all stable mixed cells in a special polyhedral subdivision called a stable mixed subdivision and creates a fine mixed subdivision for each stable mixed cell. The stable mixed subdivision depends solely on the monomial structure of P and a convex geometric lifting process. One then solves a collection of subsystems corresponding to the stable mixed cells, and uses these solutions as starting points for a finite set of nonlinear homotopies. This final set of homotopies give a dramatic computational improvement over earlier homotopy algorithms, but the initial steps of finding the stable mixed cells and their fine mixed subdivisions require many costly recursive liftings. The purpose of this paper is to present a new strategy using a single lifting which quickly (and simultaneously) builds the stable mixed subdivision, the fine mixed subdivisions of the stable mixed cells, and the necessary subsystems
Recent Advances on Determining the Number of Real Roots of Parametric Polynomials

AbstractAn explicit criterion for the determination of the numbers and multiplicities of the real/imaginary roots for polynomials with symbolic coefficients is based on a Complete Discrimination System (CDS). A CDS is a set of explicit expressions in terms of the coefficients that are sufficient for determining the numbers and multiplicities of the real and imaginary roots. Basically, the problem is considered on a total real axis and a total complex plane. However, it is often required in both practice and theory to determine the number of real roots in some interval, especially (0,∞ ) or (−∞, 0). This article is mainly devoted to solving the case in an interval, but some global results are reviewed for understanding. It is shown, with examples, how useful the CDS can be in order to understand the behaviour of the roots of an univariate polynomial in terms of the coefficients
Gröbner Bases, Invariant Theory and Equivariant Dynamics

AbstractThis paper is about algorithmic invariant theory as it is required within equivariant dynamical systems. The question of generic bifurcation equations (arbitrary equivariant polynomial vector) requires the knowledge of fundamental invariants and equivariants. We discuss computations which are related to this for finite groups and semi-simple Lie groups. We consider questions such as the completeness of invariants and equivariants. Efficient computations are gained by the Hilbert series driven Buchberger algorithm because computation of elimination ideals is heavily required. Applications such as orbit space reduction are presented
Polynomial Systems from Certain Differential Equations

AbstractIn this paper, combined elimination techniques are applied to establish relations among center conditions for certain cubic differential systems initially investigated by Kukles in 1944. The obtained relations clarify recent rediscoveries of some known conditions of Cherkas. The computational difficulties of establishing the complete center conditions for Kukles’ system, a problem that is still open, are illustrated by interactive elimination. Some generated polynomial systems that need be solved are made available electronically for other developers to test elimination algorithms and their implementations
Algorithms forq -Hypergeometric Summation in Computer Algebra

AbstractThis paper describes three algorithms for q -hypergeometric summation: • a multibasic analogue of Gosper’s algorithm, • the q - Zeilberger algorithm, and • an algorithm for finding q - hypergeometric solutions of linear recurrences together with their Maple implementations, which is relevant both to people being interested in symbolic computation and in q -series. For all these algorithms, the theoretical background is already known and has been described, so we give only short descriptions, and concentrate ourselves on introducing our corresponding Maple implementations by examples. Each section is closed with a description of the input/output specifications of the corresponding Maple command. We present applications to q -analogues of classical orthogonal polynomials. In particular, the connection coefficients between families of q -Askey–Wilson polynomials are computed. Mathematica implementations have been developed for most of these algorithms, whereas to our knowledge only Zeilberger’s algorithm has been implemented in Maple so far (Koornwinder, 1993 or Zeilberger, cf. Pe kov0sek et al., 1996). We made an effort to implement the algorithms as efficient as possible which in the q -Petkovšek case led us to an approach with equivalence classes. Hence, our implementation is considerably faster than other ones. Furthermore the q -Gosper algorithm has been generalized to also find formal power series solutions
The Fourth-order Difference Equation Satisfied by the Associated Orthogonal Polynomials of the Δ -Laguerre–Hahn Class

AbstractStarting from the Dω-Riccati difference equation satisfied by the Stieltjes function of a linear functional, we work out an algorithm which enables us to write the general fourth-order difference equation satisfied by the associated of any integer order of orthogonal polynomials of the Δ -Laguerre–Hahn class. Moreover, in classical situations (Meixner, Charlier, Krawtchouk and Hahn), we give these difference equations explicitly; and from the Hahn difference equation, by limit processes we recover the difference equations satisfied by the associated of the classical discrete orthogonal polynomials and the differential equations satisfied by the associated of the classical continuous orthogonal polynomials
Classical Invariants and 2-descent on Elliptic Curves

AbstractThe classical theory of invariants of binary quartics is applied to the problem of determining the group of rational points of an elliptic curve defined over a field K by 2-descent. The results lead to some simplifications to the method first presented in Birch and Swinnerton-Dyer (1963), and can be applied to give a more efficient algorithm for determining Mordell–Weil groups over Q, as well as being more readily extended to other number fields. In this paper we mainly restrict ourselves to general theory, valid over arbitrary fields of characteristic neither 2 nor 3
Constructions in Finite Geometry Using Computer Algebra Systems

AbstractOne way of using a computer algebra system to do research in finite geometry is to use the system to construct “small" order examples of various constructions, and then hope to recognize a pattern that can be generalized and eventually proven. Of course, initially one does not know if the “small" order examples exist. However, if one has sufficiently good insight concerning where to look and a reasonably good “starter", the computer algebra system will often find these examples quite expeditiously. Once found the system can then be used to analyze the constructs. Brute-force searching, on the other hand, is typically foolhardy with such general purpose systems. These ideas will be illustrated with two problems in finite geometry: (1) finding new translation planes by a technique called “nesting", and (2) finding large collections of pairwise disjoint projective bundles of conics
On Shortest Linear Recurrences

AbstractThis is an expository account of a constructive theorem on the shortest linear recurrences of a finite sequence over an arbitrary integral domainR. A generalization of rational approximation, which we call “realization”, plays a key role throughout the paper.We also give the associated “minimal realization” algorithm, which has a simple control structure and is division-free. It is easy to show that the number ofR-multiplications required isO(n2), wherenis the length of the input sequence.Our approach is algebraic and independent of any particular application. We view a linear recurring sequence as a torsion element in a naturalR[X]-module. The standardR[X]-module of Laurent polynomials overRunderlies our approach to finite sequences. The prerequisites are nominal and we use short Fibonacci sequences as running examples
A Categorical Critical-pair Completion Algorithm

AbstractWe introduce a general critical-pair/completion algorithm, formulated in the language of category theory. It encompasses the Knuth–Bendix procedure for term rewriting systems (also modulo equivalence relations), the Gröbner basis algorithm for polynomial ideal theory, and the resolution procedure for automated theorem proving. We show how these three procedures fit in the general algorithm, and how our approach relates to other categorical modeling approaches to these algorithms, especially term rewriting
A Computational Method for Determining Strong Stabilizability of n-D Systems

AbstractThis paper describes an algorithm for the following problem: given two multivariate complex or real polynomials f and g , decide whether there exist complex or real polynomials h and k such that both k and fh+gk have no zero in the unit polydisc. This problem, known as strong stabilizability, is fundamental in control theory, with important applications in designing stable feedback systems with a stable compensator. Our algorithm for solving the problem is formulated based on the cylindrical algebraic decomposition(cad) of an algebraic variety. While recent applications of cad to systems and control have been focused on those problems which have a quantifier elimination formulation, our method is novel in that it explicitly computes some topological properties of an algebraic variety based on the cad to solve the problem for which a quantifier elimination formulation is not readily available
Integrating Factors for Second-order ODEs

AbstractA systematic algorithm for building integrating factors of the form μ(x,y), μ(x,y′) or μ(y,y′) for second-order ODEs is presented. The algorithm can determine the existence and explicit form of the integrating factors themselves without solving any differential equations, except for a linear ODE in one subcase of the μ(x,y) problem. Examples of ODEs not having point symmetries are shown to be solvable using this algorithm. The scheme was implemented in Maple, in the framework of the ODEtools package and its ODE-solver. A comparison between this implementation and other computer algebra ODE-solvers in tackling non-linear examples from Kamke's book is shown
On the Bézout Construction of the Resultant

AbstractThis paper deals with some ideas of Bézout and his successors Poisson, Netto and Laurent for solving polynomial systems. We treat them from the determinantal and from the Gröbner basis point of view. This results in effective algorithms for constructing the multivariate resultant. Other problems of Elimination Theory are discussed: how to find an eliminant for a polynomial system, how to represent its zeroes as the rational functions of the roots of this eliminant and how to separate zeroes
A Modular Method to Compute the Rational Univariate Representation of Zero-dimensional Ideals

AbstractTo give an efficiently computable representation of the zeros of a zero-dimensional ideal I, Rouillier (1996) introduced the rational univariate representation (RUR) as an extension of the generalized shape lemma (GSL) proposed by Alonso et al. (1996). In this paper, we propose a new method to compute the RUR of the radical of I, and report on its practical implementation. In the new method, the RUR of the radical of I is computed efficiently by applying modular techniques to solving the systems of linear equations. The performance of the method is examined by practical experiments. We also discuss its theoretical efficiency
Complexity of Computing the Local Dimension of a Semialgebraic Set

AbstractThe paper describes several algorithms related to a problem of computing the local dimension of a semialgebraic set. Let a semialgebraic set V be defined by a system of k inequalities of the formf≥ 0 with f∈R [ X1,⋯ ,Xn ], deg(f) <d , andx∈V . An algorithm is constructed for computing the dimension of the Zariski tangent space to V at x in time (kd)O(n). Let x belong to a stratum of codimension lxin V with respect to a smooth stratification ofV . Another algorithm computes the local dimension dimx(V) with the complexity (k(lx+ 1)d)O(lx2n). Ifl=maxx∈Vlx, and for every connected component the local dimension is the same at each point, then the algorithm computes the dimension of every connected component with complexity (k(l+ 1)d)O(l2n). If V is a real algebraic variety defined by a system of equations, then the complexity of the algorithm is less thankdO(l2n) , and the algorithm also finds the dimension of the tangent space to V at x in time kdO(n). Whenl is fixed, like in the case of a smooth V , the complexity bounds for computing the local dimension are (kd)O(n)andkdO(n) respectively. A third algorithm finds the singular locus ofV in time (kd)O(n2)
Renormalization Automated by Hopf Algebra

AbstractIt was recently shown that the renormalization of quantum field theory is organized by the Hopf algebra of decorated rooted trees, whose coproduct identifies the divergences requiring subtraction and whose antipode achieves this. We automate this process in a few lines of recursive symbolic code, which deliver a finite renormalized expression for any Feynman diagram. We thus verify a representation of the operator product expansion, which generalizes Chen’s Lemma for iterated integrals. The subset of diagrams whose forest structure entails a unique primitive subdivergence provides a representation of the Hopf algebra HRof undecorated rooted trees. Our undecorated Hopf algebra program is designed to process the 24 213 878 BPHZ contributions to the renormalization of 7813 diagrams, with up to 12 loops. We consider 10 models, each in nine renormalization schemes. The two simplest models reveal a notable feature of the subalgebra of Connes and Moscovici, corresponding to the commutative part of the Hopf algebra HTof the diffeomorphism group: it assigns to Feynman diagrams those weights which remove zeta values from the counterterms of the minimal subtraction scheme. We devise a fast algorithm for these weights, whose squares are summed with a permutation factor, to give rational counterterms
An Optimal Algorithm for Constructing the Reduced Gröbner Basis of Binomial Ideals

AbstractIn this paper, we present an optimal, exponential space algorithm for generating the reduced Gröbner basis of binomial ideals. We make use of the close relationship between commutative semigroups and pure difference binomial ideals. Based on an optimal algorithm for the uniform word problem in commutative semigroups, we first derive an exponential space algorithm for constructing the reduced Gröbner basis of pure difference binomial ideals. In addition to some applications to finitely presented commutative semigroups, this algorithm is then extended to an exponential space algorithm for generating the reduced Gröbner basis of binomial ideals over Q in general
Practical Algorithms for Polycyclic Matrix Groups

AbstractIn this paper we describe a suite of new algorithms for studying polycyclic matrix groups—algorithms for testing membership and for uncovering the polycyclic structure of the group. We also describe an algorithm for deciding whether or not a group is solvable, which, in the important context of subgroups ofGL (n,Z), is equivalent to deciding whether or not a group is polycyclic. Algorithms were developed in Baumslag et al. (1991. The algorithmic theory of polycyclic-by-finite groups. J. Algebra, 142, 118–149) for all of these problems, but the algorithms in this paper represent a first step toward finding practical algorithms: experiments show that they are fast enough to be useful in studying some reasonably complex examples using current technology
Application of Computer Algebra Systems for Stability Analysis of Difference Schemes on Curvilinear Grids

AbstractThe paper deals with problems arising in the application of the computer algebra systems for the symbolic–numeric stability analysis of difference schemes and schemes of the finite-volume method approximating the two-dimensional Euler equations for compressible fluid flows on curvilinear spatial grids. We carry out a detailed comparison of the REDUCE 3.6 and Mathematica(Versions 2.2 and 3.0) from the point of view of their applicability to the solution of the above problems. We draw a conclusion that a preference should be given for Mathematica from the viewpoint of the execution of symbolic–numeric computations. We also describe in detail our new symbolic–numeric algorithm for stability investigation, which was implemented with the aid of Mathematica. The proposed method enables us to reduce the needed computer storage at the symbolic stages by a factor of about 20 as compared with the previous algorithms. A feature of the numerical stages is the use of the arithmetic of rational numbers, which enables us to avoid the accumulation of the roundoff errors. We present the examples of the application of the proposed symbolic–numeric method for stability analysis of very complex schemes of the finite-volume method on curvilinear grids, which are widely used in computational fluid dynamics
Generic and Cogeneric Monomial Ideals

AbstractMonomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions encoded by simplicial complexes. There are numerous equivalent ways to say that a monomial ideal is generic or cogeneric. For a generic monomial ideal, the associated primes satisfy a saturated chain condition, and the Cohen–Macaulay property implies shellability for both the Scarf complex and the Stanley–Reisner complex. Reverse lexicographic initial ideals of generic lattice ideals are generic. Cohen–Macaulayness for cogeneric ideals is characterized combinatorially; in the cogeneric case, the Cohen–Macaulay type is greater than or equal to the number of irreducible components. Methods of proof include Alexander duality and Stanley’s theory of local h -vectors
An Algorithmic Approach to Fundamental Groups and Covers of Combinatorial Cell Complexes

AbstractWe first develop a construction, originally due to Reidemeister, of the fundamental group and covers of a two-dimensional combinatorial cell complex. Then, we describe a practical algorithmic approach to the computation of fundamental groups and first homology groups (as finitely presented groups), of first homology groups mod p(as vector spaces), of deck groups (as permutation groups), and of covers of finite simple such complexes. In the case of clique complexes of finite simple graphs, the algorithms described have been implemented in GAP, making use of the GRAPE package
Using Rewriting Systems to Compute Left Kan Extensions and Induced Actions of Categories

AbstractThe aim is to apply string-rewriting methods to compute left Kan extensions, or, equivalently, induced actions of monoids, categories, groups or groupoids. This allows rewriting methods to be applied to a greater range of situations and examples than before. The data for the rewriting is called a Kan extension presentation. The paper has its origins in earlier work by Carmody and Walters who gave an algorithm for computing left Kan extensions based on extending the Todd–Coxeter procedure, an algorithm only applicable when the induced action is finite. The current work, in contrast, gives information even when the induced action is infinite
Toric Newton Method for Polynomial Homotopies

AbstractThis paper defines a generalization of Newton’s method to deal with solution paths defined by polynomial homotopies that lead to extremal values. Embedding the solutions in a toric variety leads to explicit scaling relations between coefficients and solutions. Toric Newton is a symbolic-numeric algorithm where the symbolic pre-processing exploits the polyhedral structures. The numerical stage uses the additional variables introduced by the homogenization to scale the components of the solution vectors to the complex unit circle. Toric Newton generates appropriate affine charts and enables one to approximate the magnitude of large solutions of polynomial systems
Local Cohomology at Monomial Ideals

AbstractWe prove that if B⊂R=k [ X1,⋯ , Xn] is a reduced monomial ideal, then HBi(R) = ∪d≥1ExtRi(R/B[ d ], R), where B[ d ]is the d th Frobenius power of B. We give two descriptions for HBi(R) in each multidegree, as simplicial cohomology groups of certain simplicial complexes. As a first consequence, we derive a relation between ExtR(R/B, R) and TorR(B∨, k), where B∨is the Alexander dual of B. As a further application, we give a filtration of ExtRi(R/B, R) such that the quotients are suitable shifts of modules of the form R/(Xi1,⋯ , Xir). We conclude by giving a topological description of the associated primes of ExtRi(R/B, R). In particular, we characterize the minimal associated primes of ExtRi(R/B, R) using only the Betti numbers of B∨
Linear Problems in Valued Fields

AbstractA first-order formula over a valued field is called linear if it contains no products or reciprocals of quantified variables. We give quantifier elimination procedures based on test term ideas for linear formulas in the following classes of valued fields: discretely valued fields, discretely valued fields with a Z -group as the value group over a language containing predicates stating divisibility in the value group, and non-discretely valued fields. From the existence of the elimination procedures, it follows that the corresponding decision problems are in an alternating single exponential time-space (Berman) complexity class. We exhibit the substructure completeness of the considered classes of valued fields w.r.t. linear formulas
Generalized Resultants over Unirational Algebraic Varieties

AbstractIn this paper, we propose a new method, based on Bezoutian matrices, for computing a nontrivial multiple of the resultant over a projective variety X, which is described on an open subset by a parameterization. This construction, which generalizes the classical and toric one, also applies for instance to blowing up varieties and to residual intersection problems. We recall the classical notion of resultant over a variety X. Then we extend it to varieties which are parameterized on a dense open subset and give new conditions for the existence of the resultant over these varieties. We prove that any maximal nonzero minor of the corresponding Bezoutian matrix yields a nontrivial multiple of the resultant. We end with some experiments