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  • Families of blowups of the real affine plane: Classification, isotopies and visualization
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-10-19
    Markus Brodmann; Peter Schenzel

    We classify embedded blowups of the real affine plane up to oriented isomorphy. We show that two blowups in the same isomorphism class are isotopic, using a matrix deformation argument similar to an idea given in Shastri (2002). This answers two questions which were motivated by the interactive visualizations of such blowups (see Schenzel and Stussak (2013), Stussak (2007), Stussak).

    更新日期:2020-10-19
  • Fast transforms over finite fields of characteristic two
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-10-16
    Nicholas Coxon

    We describe new fast algorithms for evaluation and interpolation on the “novel” polynomial basis over finite fields of characteristic two introduced by Lin, Chung and Han (FOCS 2014). Fast algorithms are also described for converting between their basis and the monomial basis, as well as for converting to and from the Newton basis associated with the evaluation points of the evaluation and interpolation

    更新日期:2020-10-17
  • Efficiently factoring polynomials modulo p4
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-10-15
    Ashish Dwivedi; Rajat Mittal; Nitin Saxena

    Polynomial factoring has famous practical algorithms over fields– finite, rational and p-adic. However, modulo prime powers, factoring gets harder because there is non-unique factorization and a combinatorial blowup ensues. For example, x2+pmodp2 is irreducible, but x2+pxmodp2 has exponentially many factors in the input size (which here is logarithmic in p)! We present the first randomized poly(deg⁡f

    更新日期:2020-10-16
  • Solving determinantal systems using homotopy techniques
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-09-30
    Jon D. Hauenstein; Mohab Safey El Din; Éric Schost; Thi Xuan Vu

    Let K be a field of characteristic zero and let K‾ be an algebraic closure of K. Consider a sequence of polynomials G=(g1,…,gs) in K[X1,…,Xn] with s

    更新日期:2020-09-30
  • Computing invariants for multipersistence via spectral systems and effective homology
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-09-29
    Andrea Guidolin; Jose Divasón; Ana Romero; Francesco Vaccarino

    Both spectral sequences and persistent homology are tools in algebraic topology defined from filtrations of objects (e.g. topological spaces or simplicial complexes) indexed over the set Z of integer numbers. A recent work has shown the details of the relation between both concepts. Moreover, generalizations of both concepts have been proposed which originate from a different choice of the set of indices

    更新日期:2020-09-30
  • Catalan-many tropical morphisms to trees; Part I: Constructions
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-09-25
    Jan Draisma; Alejandro Vargas

    We investigate the tree gonality of a genus-g metric graph, defined as the minimum degree of a tropical morphism from any tropical modification of the metric graph to a metric tree. We give a combinatorial constructive proof that this number is at most ⌈g/2⌉+1, a fact whose proofs so far required an algebro-geometric detour via special divisors on curves. For even genus, the tropical morphism which

    更新日期:2020-09-25
  • Standard monomial theory and toric degenerations of Schubert varieties from matching field tableaux
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-09-24
    Oliver Clarke; Fatemeh Mohammadi

    We study Gröbner degenerations of Schubert varieties inside flag varieties. We consider toric degenerations of flag varieties induced by matching fields and semi-standard Young tableaux. We describe an analogue of matching field ideals for Schubert varieties inside the flag variety and give a complete characterization of toric ideals among them. We use a combinatorial approach to standard monomial

    更新日期:2020-09-25
  • Distance to the stochastic part of phylogenetic varieties
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-09-22
    Marta Casanellas; Jesús Fernández-Sánchez; Marina Garrote-López

    Modelling the substitution of nucleotides along a phylogenetic tree is usually done by a hidden Markov process. This allows to define a distribution of characters at the leaves of the trees and one might be able to obtain polynomial relationships among the probabilities of different characters. The study of these polynomials and the geometry of the algebraic varieties defined by them can be used to

    更新日期:2020-09-22
  • Combinatorial decompositions for monomial ideals
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-09-22
    Michela Ceria

    It is well known that Riquier introduced the notion of multiplicative variables applied to order ideals to represent intitial conditions of partial differential equations as series. On the other hands, Janet introduced the concept of involutive division. Following Riquier and Janet, we focus on the following problem. Suppose one needs to compute a monomial ideal generated in some degree D and its escalier

    更新日期:2020-09-22
  • On Symbolic Integration of Algebraic Functions
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-09-11
    M.D. Malykh; L.A. Sevastianov; Y. Yu

    Algorithms for integration of the algebraic functions implemented in modern computer algebra systems (CAS) are not always able to solve the classical problems of integration in the class of algebraic or elementary functions. The most general approach to describing the integral of an algebraic function is to find a standard representation for Abelian integrals, which, on the one hand, would not be too

    更新日期:2020-09-11
  • Unirational differential curves and differential rational parametrizations
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-09-03
    Lei Fu; Wei Li

    In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential curves. Then we show that the implicitization problem of proper linear differential rational parametric equations can be solved by means of differential resultants. Furthermore

    更新日期:2020-09-03
  • An Algorithmic approach to small limit cycles of nonlinear differential systems: The averaging method revisited
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-09-02
    Bo Huang; Chee Yap

    This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging method. The first algorithm allows one to transform the considered differential systems to the normal form of averaging. Here, we restricted the unperturbed term of

    更新日期:2020-09-02
  • A condition for multiplicity structure of univariate polynomials
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-08-26
    Hoon Hong; Jing Yang

    We consider the problem of finding a condition for a univariate polynomial having a given multiplicity structure when the number of distinct roots is given. It is well known that such conditions can be written as conjunctions of several polynomial equations and one inequation in the coefficients, by using repeated parametric gcd's. In this paper, we give a novel condition which is not based on repeated

    更新日期:2020-08-26
  • On the existence of telescopers for rational functions in three variables
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-08-20
    Shaoshi Chen; Lixin Du; Rong-Hua Wang; Chaochao Zhu

    Zeilberger's method of creative telescoping is crucial for the computer-generated proofs of combinatorial and special-function identities. Telescopers are linear differential or (q-)recurrence operators computed by algorithms for creative telescoping. Two fundamental problems related to creative telescoping are whether telescopers exist, and how to construct them efficiently when they do. In this paper

    更新日期:2020-08-20
  • Exact p-adic computation in Magma
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-08-17
    Christopher Doris

    We describe a new arithmetic system for the Magma computer algebra system for working with p-adic numbers exactly, in the sense that numbers are represented lazily to infinite p-adic precision. This is the first highly featured such implementation. This has the benefits of increasing user-friendliness and speeding up some computations, as well as forcibly producing provable results. We give theoretical

    更新日期:2020-08-17
  • Polynomial-time proofs that groups are hyperbolic
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-08-14
    Derek Holt; Stephen Linton; Max Neunhöffer; Richard Parker; Markus Pfeiffer; Colva M. Roney-Dougal

    It is undecidable in general whether a given finitely presented group is word hyperbolic. We use the concept of pregroups, introduced by Stallings in Stallings (1971), to define a new class of van Kampen diagrams, which represent groups as quotients of virtually free groups. We then present a polynomial-time procedure that analyses these diagrams, and either returns an explicit linear Dehn function

    更新日期:2020-08-14
  • Approximate square-free part and decomposition
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-08-13
    Kosaku Nagasaka

    Square-free decomposition is one of fundamental computations for polynomials. However, any conventional algorithm may not work for polynomials with a priori errors on their coefficients. There are mainly two approaches to overcome this empirical situation: approximate polynomial GCD (greatest common divisor) and the nearest singular polynomial. In this paper, we show that these known approaches are

    更新日期:2020-08-14
  • Reconstruction of rational ruled surfaces from their silhouettes
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-08-10
    Matteo Gallet; Niels Lubbes; Josef Schicho; Jan Vršek

    We provide algorithms to reconstruct rational ruled surfaces in three-dimensional projective space from the “apparent contour” of a single projection to the projective plane. We deal with the case of tangent developables and of general projections to P3 of rational normal scrolls. In the first case, we use the fact that every such surface is the projection of the tangent developable of a rational normal

    更新日期:2020-08-10
  • A fast algorithm for computing multiplicative relations between the roots of a generic polynomial
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-08-06
    Tao Zheng

    Multiplicative relations between the roots of a polynomial in Q[x] have drawn much attention in the field of arithmetic and algebra, while the problem of computing these relations is interesting to researchers in many other fields. In this paper, a sufficient condition is given for a polynomial f∈Q[x] to have only trivial multiplicative relations between its roots, which is a generalization of those

    更新日期:2020-08-06
  • Reducing radicals in the spirit of Euclid
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-15
    Kurt Girstmair

    Let p be an odd natural number ≥3. Inspired by results from Euclid's Elements, we express the irrationaly=d+Rp, whose degree is 2p, as a polynomial function of irrationals of degrees ≤p. In certain cases y is expressed by simple radicals. This reduction of the degree exhibits remarkably regular patterns of the polynomials involved. The proof is based on hypergeometric summation, in particular, on Zeilberger's

    更新日期:2020-07-15
  • Catalecticant intersections and confinement of decompositions of forms
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-09
    Elena Angelini; Cristiano Bocci; Luca Chiantini

    We introduce the notion of confinement of decompositions for forms or vector of forms. The confinement, when it holds, lowers the number of parameters that one needs to consider, in order to find all the possible decompositions of a given set of data. With the technique of confinement, we obtain here two results. First, we give a new, shorter proof of a result by London (1890) that 3 general plane

    更新日期:2020-07-10
  • A new general formula for the Cauchy Index on an interval with Subresultants
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-08
    Daniel Perrucci; Marie-Françoise Roy

    We present a new formula for the Cauchy index of a rational function on an interval using subresultant polynomials. There is no condition on the endpoints of the interval and the formula also involves in some cases less subresultant polynomials.

    更新日期:2020-07-08
  • Strict inclusions of high rank loci
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-08
    Edoardo Ballico; Alessandra Bernardi; Emanuele Ventura

    For a given projective variety X, the high rank loci are the closures of the sets of points whose X-rank is higher than the generic one. We show examples of strict inclusion arising from two consecutive high rank loci. Our first example comes from looking at the Veronese surface of plane quartics. Although Piene had already shown an example in which X is a curve, we construct infinitely many curves

    更新日期:2020-07-08
  • Computing Quotients by Connected Solvable Groups
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-08
    Gregor Kemper

    Consider an action of a connected solvable group G on an affine variety X. This paper presents an algorithm that constructs a semi-invariant f∈K[X]=:R and computes the invariant ring (Rf)G together with a presentation. The morphism Xf→Spec((Rf)G) obtained from the algorithm is a universal geometric quotient. In fact, it is even better than that: a so-called excellent quotient. If R is a polynomial

    更新日期:2020-07-08
  • Voronoi cells of varieties
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-08
    Diego Cifuentes; Kristian Ranestad; Bernd Sturmfels; Madeleine Weinstein

    Every real algebraic variety determines a Voronoi decomposition of its ambient Euclidean space. Each Voronoi cell is a convex semialgebraic set in the normal space of the variety at a point. We compute the algebraic boundaries of these Voronoi cells.

    更新日期:2020-07-08
  • Quartic monoid surfaces with maximum number of lines
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-08
    Mauro Carlo Beltrametti; Alessandro Logar; Maria-Laura Torrente

    In 1884 the German mathematician Karl Rohn published a substantial paper Rohn (1884) on the properties of quartic surfaces with triple points, proving (among many other things) that the maximum number of lines contained in a quartic monoid surface is 31. In this paper we study in details this class of surfaces. We prove that there exists an open subset A⊆PK1 (K is a characteristic zero field) that

    更新日期:2020-07-08
  • Autocovariance varieties of moving average random fields
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-08
    Carlos Améndola; Viet Son Pham

    We study the autocovariance functions of moving average random fields over the integer lattice Zd from an algebraic perspective. These autocovariances are parametrized polynomially by the moving average coefficients, hence tracing out algebraic varieties. We derive dimension and degree of these varieties and we use their algebraic properties to obtain statistical consequences such as identifiability

    更新日期:2020-07-08
  • On certain polynomial systems involving Stirling numbers of second kind
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-08
    F.J. Castro-Jiménez; H. Cobo Pablos

    We solve a special type of linear systems with coefficients in multivariate polynomial rings. These systems arise in the computation of b-functions with respect to weights of certain hypergeometric ideals in the Weyl algebra.

    更新日期:2020-07-08
  • Initial Steps in the Classification of Maximal Mediated Sets
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-08
    Jacob Hartzer; Olivia Röhrig; Timo de Wolff; Oguzhan Yürük

    Maximal mediated sets (MMS), introduced by Reznick, are distinguished subsets of lattice points in integral polytopes with even vertices. MMS of Newton polytopes of AGI-forms and nonnegative circuit polynomials determine whether these polynomials are sums of squares. In this article, we take initial steps in classifying MMS both theoretically and practically. Theoretically, we show that MMS of simplices

    更新日期:2020-07-08
  • Stronger bounds on the cost of computing Gröbner bases for HFE systems
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-08
    Elisa Gorla; Daniela Mueller; Christophe Petit

    We give upper bounds for the solving degree and the last fall degree of the polynomial system associated to the HFE (Hidden Field Equations) cryptosystem. Our bounds improve the known bounds for this type of systems. We also present new results on the connection between the solving degree and the last fall degree and prove that, in some cases, the solving degree is independent of coordinate changes

    更新日期:2020-07-08
  • Unexpected hypersurfaces with multiple fat points
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-08
    Justyna Szpond

    Starting with the ground-breaking work of Cook II, Harbourne, Migliore and Nagel, there has been a lot of interest in unexpected hypersurfaces. In the last couple of months a considerable number of new examples and new phenomena has been observed and reported on. All examples studied so far had just one fat point. In this note we introduce a new series of examples, which establishes for the first time

    更新日期:2020-07-08
  • Saturations of Subalgebras, SAGBI Bases, and U-invariants
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-08
    Anna Maria Bigatti; Lorenzo Robbiano

    Given a polynomial ring P over a field K, an element g∈P, and a K-subalgebra S of P, we deal with the problem of saturating S with respect to g, i.e. computing Satg(S)=S[g,g−1]∩P. In the general case we describe a procedure/algorithm to compute a set of generators for Satg(S) which terminates if and only if it is finitely generated. Then we consider the more interesting case when S is graded. In particular

    更新日期:2020-07-08
  • Computing integral bases via localization and Hensel lifting
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-08
    Janko Böhm; Wolfram Decker; Santiago Laplagne; Gerhard Pfister

    We present a new algorithm for computing integral bases in algebraic function fields of one variable, or equivalently for constructing the normalization of a plane curve. Our basic strategy makes use of the concepts of localization and completion, together with the Chinese remainder theorem, to reduce the problem to the task of finding integral bases for the branches of each singularity of the curve

    更新日期:2020-07-08
  • Certification for polynomial systems via square subsystems
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-08
    Timothy Duff; Nickolas Hein; Frank Sottile

    We consider numerical certification of approximate solutions to a system of polynomial equations with more equations than unknowns by first certifying solutions to a square subsystem. We give several approaches that certifiably select which are solutions to the original overdetermined system. These approaches each use different additional information for this certification, such as liaison, Newton-Okounkov

    更新日期:2020-07-08
  • Computing representation matrices for the Frobenius on cohomology groups
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-07
    Momonari Kudo

    In algebraic geometry, the Frobenius map F⁎ on cohomology groups play an important role in the classification of algebraic varieties over a field of positive characteristic. In particular, representation matrices for F⁎ give rise to many important invariants such as p-rank and a-number. Several methods for computing representation matrices for F⁎ have been proposed for specific curves. In this paper

    更新日期:2020-07-07
  • Measuring the local non-convexity of real algebraic curves
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-07
    Miruna-Ştefana Sorea

    The goal of this paper is to measure the non-convexity of compact and smooth connected components of real algebraic plane curves. We study these curves first in a general setting and then in an asymptotic one. In particular, we consider sufficiently small levels of a real bivariate polynomial in a small enough neighbourhood of a strict local minimum at the origin of the real affine plane. We introduce

    更新日期:2020-07-07
  • On a tropical version of the Jacobian conjecture
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-07
    Dima Grigoriev; Danylo Radchenko

    We prove that, for a tropical rational map if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical polynomial map on the plane is an isomorphism if all the Jacobians have the same sign (positive or negative). In addition, for a tropical rational map we prove

    更新日期:2020-07-07
  • A constructive method for decomposing real representations
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-07-04
    Sajid Ali; Hassan Azad; Indranil Biswas; Willem A. de Graaf

    A constructive method for decomposing finite dimensional representations of semisimple real Lie algebras is developed. The method is illustrated by an example. We also discuss an implementation of the algorithm in the language of the computer algebra system GAP4.

    更新日期:2020-07-06
  • Computing strong regular characteristic pairs with Gröbner bases
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-06-25
    Rina Dong; Dongming Wang

    The W-characteristic set of a polynomial ideal is the minimal triangular set contained in the reduced lexicographical Gröbner basis of the ideal. A pair (G,C) of polynomial sets is a strong regular characteristic pair if G is a reduced lexicographical Gröbner basis, C is the W-characteristic set of the ideal 〈G〉, the saturated ideal sat(C) of C is equal to 〈G〉, and C is regular. In this paper, we show

    更新日期:2020-06-25
  • Singularities and Genus of the k-Ellipse
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-06-25
    Yuhan Jiang; Weiqiao Han

    A k-ellipse is a plane curve consisting of all points whose distances from k fixed foci sum to a constant. We determine the singularities and genus of its Zariski closure in the complex projective plane. The paper resolves an open problem stated by Nie, Parrilo and Sturmfels in 2008.

    更新日期:2020-06-25
  • Existence and convergence of Puiseux series solutions for autonomous first order differential equations
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-06-25
    José Cano; Sebastian Falkensteiner; J.Rafael Sendra

    Given an autonomous first order algebraic ordinary differential equation F(y,y′)=0, we prove that every formal Puiseux series solution of F(y,y′)=0, expanded around any finite point or at infinity, is convergent. The proof is constructive and we provide an algorithm to describe all such Puiseux series solutions. Moreover, we show that for any point in the complex plane there exists a solution of the

    更新日期:2020-06-25
  • Multilinear Polynomial Systems: Root Isolation and Bit Complexity
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-06-18
    Ioannis Z. Emiris; Angelos Mantzaflaris; Elias P. Tsigaridas

    We exploit structure in polynomial system solving by considering polynomials that are linear in subsets of the variables. We focus on algorithms and their Boolean complexity for computing isolating hyperboxes for all the isolated complex roots of well-constrained, unmixed systems of multilinear polynomials based on resultant methods. We enumerate all expressions of the multihomogeneous (or multigraded)

    更新日期:2020-06-18
  • A nearly optimal algorithm to decompose binary forms
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-06-18
    Matías R. Bender; Jean-Charles Faugère; Ludovic Perret; Elias Tsigaridas

    Symmetric tensor decomposition is an important problem with applications in several areas, for example signal processing, statistics, data analysis and computational neuroscience. It is equivalent to Waring's problem for homogeneous polynomials, that is to write a homogeneous polynomial in n variables of degree D as a sum of D-th powers of linear forms, using the minimal number of summands. This minimal

    更新日期:2020-06-18
  • Verification Protocols with Sub-Linear Communication for Polynomial Matrix Operations
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-06-18
    David Lucas; Vincent Neiger; Clément Pernet; Daniel S. Roche; Johan Rosenkilde

    We design and analyze new protocols to verify the correctness of various computations on matrices over the ring F[x] of univariate polynomials over a field F. For the sake of efficiency, and because many of the properties we verify are specific to matrices over a principal ideal domain, we cannot simply rely on previously-developed linear algebra protocols for matrices over a field. Our protocols are

    更新日期:2020-06-18
  • Counting invariant subspaces and decompositions of additive polynomials
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-06-18
    Joachim von zur Gathen; Mark Giesbrecht; Konstantin Ziegler

    The functional (de)composition of polynomials is a topic in pure and computer algebra with many applications. The structure of decompositions of (suitably normalized) polynomials f=g∘h in F[x] over a field F is well understood in many cases, but less well when the degree of f is divisible by the positive characteristic p of F. This work investigates the decompositions of r-additive polynomials, where

    更新日期:2020-06-18
  • Toward the best algorithm for approximate GCD of univariate polynomials
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-06-17
    Kosaku Nagasaka

    Approximate polynomial GCD (greatest common divisor) of polynomials with a priori errors on their coefficients, is one of interesting problems in Symbolic-Numeric Computations. In fact, there are many known algorithms: QRGCD, UVGCD, STLN based methods, Fastgcd and so on. The fundamental question of this paper is “which is the best?” from the practical point of view, and subsequently “is there any better

    更新日期:2020-06-17
  • An Unwinding Number Pair for Continuous Expressions of Integrals
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-06-17
    Robert H.C. Moir; Robert M. Corless; David J. Jeffrey

    We consider the problem of obtaining expressions for integrals that are continuous over the entire domain of integration where the true mathematical integral is continuous, which has been called the problem of obtaining integrals on domains of maximum extent (Jeffrey, 1993). We develop a method for correcting discontinuous integrals using an extension of the concept of unwinding numbers for complex

    更新日期:2020-06-17
  • Drinfeld modules with complex multiplication, Hasse invariants and factoring polynomials over finite fields
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-06-17
    Javad Doliskani; Anand Kumar Narayanan; Éric Schost

    We present a novel randomized algorithm to factor polynomials over a finite field Fq of odd characteristic using rank 2 Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial f∈Fq[x] to be factored) with respect to a random Drinfeld module ϕ with complex multiplication. Factors of f supported on prime ideals with supersingular

    更新日期:2020-06-17
  • A fast parallel sparse polynomial GCD algorithm
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-06-17
    Jiaxiong Hu; Michael Monagan

    We present a parallel GCD algorithm for sparse multivariate polynomials with integer coefficients. The algorithm combines a Kronecker substitution with a Ben-Or/Tiwari sparse interpolation modulo a smooth prime to determine the support of the GCD. We have implemented our algorithm in C for primes of various size and have parallelized it using Cilk C. We compare our implementation with Maple and Magma's

    更新日期:2020-06-17
  • Symbolic analysis of multiple steady states in a MAPK chemical reaction network
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-06-17
    Daniel Lichtblau

    We consider the problem of analyzing chemical reaction networks that may allow multiple positive steady states. We use tools from “classical” computer algebra (Gröbner bases over a parametrized domain, computation of a discriminant variety, graphical and mathematical analysis of solution sets, cylindrical decomposition) to help determine regions of stoichiometric compatibility classes that have multiple

    更新日期:2020-06-17
  • On some classes of irreducible polynomials
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-06-17
    Jaime Gutierrez; Jorge Jiménez Urroz

    One of the fundamental tasks of Symbolic Computation is the factorization of polynomials into irreducible factors. The aim of the paper is to produce new families of irreducible polynomials, generalizing previous results in the area. One example of our general result is that for a near-separated polynomial, i.e., polynomials of the form F(x,y)=f1(x)f2(y)−f2(x)f1(y), then F(x,y)+r is always irreducible

    更新日期:2020-06-17
  • On the Computation of Identities Relating Partition Numbers in Arithmetic Progressions with Eta Quotients: An Implementation of Radu's Algorithm
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-05-21
    Nicolas Allen Smoot

    In 2015 Cristian-Silviu Radu designed an algorithm to detect identities of a class studied by Ramanujan and Kolberg, in which the generating functions of a partition function over a given set of arithmetic progression are expressed in terms of Dedekind eta quotients over a given congruence subgroup. These identities include the famous results by Ramanujan which provide a witness to the divisibility

    更新日期:2020-05-21
  • Computing Unit Groups of Curves
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-05-11
    Justin Chen; Sameera Vemulapalli; Leon Zhang

    The group of units modulo constants of an affine variety over an algebraically closed field is free abelian of finite rank. Computing this group is difficult but of fundamental importance in tropical geometry, where it is necessary in order to realize intrinsic tropicalizations. We present practical algorithms for computing unit groups of smooth curves of low genus. Our approach is rooted in divisor

    更新日期:2020-05-11
  • Distance invariant method for normalization of indexed differentials
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-05-11
    Jiang Liu; Feng Ni

    A distance from free to dummy indices is defined. The distance is invariant with respect to both monoterm symmetries and bottom antisymmetry. Using the distance invariant, we present an index-replacement algorithm. We then develop two normalization algorithms. One is with respect to monoterm symmetries and has complexity lower than known algorithms; the other allows one to determine the equivalence

    更新日期:2020-05-11
  • The dimension of the moduli spaces of curves defined by topologically non quasi-homogeneous functions
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-05-04
    Jinan Loubani

    We consider a topological class of a germ of complex analytic function in two variables which does not belong to its jacobian ideal. Such a function is not quasi homogeneous. The 0-level of such a function defines a germ of analytic curve. Proceeding similarly to the homogeneous case Genzmer and Paul (2011) and the quasi homogeneous case Genzmer and Paul (2016), we describe an algorithm which computes

    更新日期:2020-05-04
  • Computing the multilinear factors of lacunary polynomials without heights
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-04-30
    Arkadev Chattopadhyay; Bruno Grenet; Pascal Koiran; Natacha Portier; Yann Strozecki

    We present a deterministic algorithm which computes the multilinear factors of multivariate lacunary polynomials over number fields. Its complexity is polynomial in ℓn where ℓ is the lacunary size of the input polynomial and n its number of variables, that is in particular polynomial in the logarithm of its degree. We also provide a randomized algorithm for the same problem of complexity polynomial

    更新日期:2020-04-30
  • Symmetric polynomials in upper-bound semirings
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-02-07
    Sara Kališnik; Davorin Lešnik

    The fundamental theorem of symmetric polynomials over rings is a classical result which states that every unital commutative ring is elementary, i.e. we can express symmetric polynomials with elementary ones in a unique way. The result does not extend directly to polynomials over semirings, but we do have analogous results for some special semirings, for example, the tropical, extended and supertropical

    更新日期:2020-02-07
  • Effective coefficient asymptotics of multivariate rational functions via semi-numerical algorithms for polynomial systems
    J. Symb. Comput. (IF 0.673) Pub Date : 2020-01-22
    Stephen Melczer; Bruno Salvy

    The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several variables (ACSV) makes it possible to compute asymptotic expansions. We consider these methods from the point of view of effectivity. In particular, given a rational

    更新日期:2020-01-22
  • A generalization of the Katzman-Zhang algorithm
    J. Symb. Comput. (IF 0.673) Pub Date : 2019-12-27
    Mehmet Yeşil

    In this paper, we study the notion of special ideals. We generalize the results on those as well as the algorithm obtained for finite dimensional power series rings by Mordechai Katzman and Wenliang Zhang to finite dimensional polynomial rings.

    更新日期:2019-12-27
  • Lexicographic and reverse lexicographic quadratic Gröbner bases of cut ideals
    J. Symb. Comput. (IF 0.673) Pub Date : 2019-12-19
    Ryuichi Sakamoto

    Hibi conjectured that if a toric ideal has a quadratic Gröbner basis, then the toric ideal has either a lexicographic or a reverse lexicographic quadratic Gröbner basis. In this paper, we present a cut ideal of a graph that serves as a counterexample to this conjecture. We also discuss the existence of a quadratic Gröbner basis of a cut ideal of a cycle. Nagel and Petrović claimed that a cut ideal

    更新日期:2019-12-19
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