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On exact Reznick, Hilbert-Artin and Putinar's representations J. Symb. Comput. (IF 0.673) Pub Date : 2021-04-01 Victor Magron, Mohab Safey El Din
We consider the problem of computing exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers. We provide a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions with rational coefficients for polynomials lying in the interior of the SOS cone. The first step of this algorithm computes
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A new algorithm for computing logarithmic vector fields along an isolated singularity and Bruce-Roberts Milnor ideals J. Symb. Comput. (IF 0.673) Pub Date : 2021-03-26 Katsusuke Nabeshima, Shinichi Tajima
A new algorithm is introduced for computing logarithmic vector fields along a hypersurface with an isolated singularity. The key ideas of the algorithm are computing an ideal quotient in a polynomial ring and the use of algebraic local cohomology. The use of these ideas allows us to compute a module, over a local ring, of germs logarithmic vector fields. The resulting algorithms are much faster than
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Semialgebraic sets and real binary forms decompositions J. Symb. Comput. (IF 0.673) Pub Date : 2021-03-17 M. Ansola, A. Díaz-Cano, M.A. Zurro
The Waring Problem over polynomial rings asks how to decompose a homogeneous polynomial p of degree d as a linear combination of d-th powers of linear forms. In this work we give an algorithm to obtain a real Waring decomposition of any given real binary form p of length at most its degree. In fact, we construct a semialgebraic family of Waring decompositions for p. We illustrate our results with some
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Log-concavity of P-recursive sequences J. Symb. Comput. (IF 0.673) Pub Date : 2021-03-29 Qing-hu Hou, Guojie Li
We consider the higher order Turán inequality and higher order log-concavity for sequences {an}n≥0 such thatan−1an+1an2=1+∑i=1mri(logn)nαi+o(1nβ), where m is a nonnegative integer, αi are real numbers, ri(x) are rational functions of x and0<α1<α2<⋯<αm<β. We will give a sufficient condition on the higher order Turán inequality and the ℓ-log-concavity for n sufficiently large. Many P-recursive sequences
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Construction of free differential algebras by extending Gröbner-Shirshov bases J. Symb. Comput. (IF 0.673) Pub Date : 2021-03-17 Yunnan Li, Li Guo
As a fundamental notion, the free differential algebra on a set is concretely constructed as the polynomial algebra on the differential variables. Such a construction is not known for the more general notion of the free differential algebra on an algebra, from the left adjoint functor of the forgetful functor from differential algebras to algebras, instead of sets. In this paper we show that a generator-relation
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qFunctions – A Mathematica package for q-series and partition theory applications J. Symb. Comput. (IF 0.673) Pub Date : 2021-03-01 Jakob Ablinger, Ali Kemal Uncu
We describe the qFunctions Mathematica package for q-series and partition theory applications. This package includes both experimental and symbolic tools. The experimental set of elements includes guessers for q-difference equations and recurrences for given q-difference and fitting/finding explicit expressions for sequences of polynomials. This package can symbolically handle formal manipulations
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Efficient q-integer linear decomposition of multivariate polynomials J. Symb. Comput. (IF 0.673) Pub Date : 2021-02-18 Mark Giesbrecht, Hui Huang, George Labahn, Eugene Zima
We present two new algorithms for the computation of the q-integer linear decomposition of a multivariate polynomial. Such a decomposition is essential for the treatment of q-hypergeometric symbolic summation via creative telescoping and for describing the q-counterpart of Ore-Sato theory. Both of our algorithms require only basic integer and polynomial arithmetic and work for any unique factorization
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Symmetric ideals, Specht polynomials and solutions to symmetric systems of equations J. Symb. Comput. (IF 0.673) Pub Date : 2021-02-18 Philippe Moustrou, Cordian Riener, Hugues Verdure
An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the leading monomials of polynomials in the ideal and the Specht polynomials contained in the ideal. This provides applications in several contexts. Most notably, this
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Towards a computational prSoof of Vizing's conjecture using semidefinite programming and sums-of-squares J. Symb. Comput. (IF 0.673) Pub Date : 2021-02-01 Elisabeth Gaar, Daniel Krenn, Susan Margulies, Angelika Wiegele
Vizing's conjecture (open since 1968) relates the product of the domination numbers of two graphs to the domination number of their Cartesian product graph. In this paper, we formulate Vizing's conjecture as a Positivstellensatz existence question. In particular, we select classes of graphs according to their number of vertices and their domination number and encode the conjecture as an ideal/polynomial
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On rational and hypergeometric solutions of linear ordinary difference equations in ΠΣ⁎-field extensions J. Symb. Comput. (IF 0.673) Pub Date : 2021-02-02 Sergei A. Abramov, Manuel Bronstein, Marko Petkovšek, Carsten Schneider
We present a complete algorithm that computes all hypergeometric solutions of homogeneous linear difference equations and rational solutions of parameterized linear difference equations in the setting of ΠΣ⁎-fields. More generally, we provide a flexible framework for a big class of difference fields that are built by a tower of ΠΣ⁎-field extensions over a difference field that enjoys certain algorithmic
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Multivariate interpolation: Preserving and exploiting symmetry J. Symb. Comput. (IF 0.673) Pub Date : 2021-01-29 Erick Rodriguez Bazan, Evelyne Hubert
Interpolation is a prime tool in algebraic computation while symmetry is a qualitative feature that can be more relevant to a mathematical model than the numerical accuracy of the parameters. The article shows how to exactly preserve symmetry in multivariate interpolation while exploiting it to alleviate the computational cost. We revisit minimal degree and least interpolation with symmetry adapted
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Variadic equational matching in associative and commutative theories J. Symb. Comput. (IF 0.673) Pub Date : 2021-01-19 Besik Dundua, Temur Kutsia, Mircea Marin
In this paper we study matching in equational theories that specify counterparts of associativity and commutativity for variadic function symbols. We design a procedure to solve a system of matching equations and prove its termination, soundness, completeness, and minimality. The minimal complete set of matchers for such a system can be infinite, but our algorithm computes its finite representation
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Computing the Schur multipliers of the Lie p-rings in the family defined by a symbolic Lie p-ring presentation J. Symb. Comput. (IF 0.673) Pub Date : 2021-01-05 Bettina Eick, Taleea Jalaeeyan Ghorbanzadeh
A symbolic Lie p-ring presentation defines a family of nilpotent Lie rings with pn elements for infinitely many primes p and a fixed positive integer n. Symbolic Lie p-ring presentations are used in the classification of isomorphism types of nilpotent Lie rings of order pn for all primes p and n≤7. We describe an algorithm to compute the Schur multipliers of all nilpotent Lie rings in the family defined
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Rationalizability of square roots J. Symb. Comput. (IF 0.673) Pub Date : 2021-01-04 Marco Besier, Dino Festi
Feynman integral computations in theoretical high energy particle physics frequently involve square roots in the kinematic variables. Physicists often want to solve Feynman integrals in terms of multiple polylogarithms. One way to obtain a solution in terms of these functions is to rationalize all occurring square roots by a suitable variable change. In this paper, we give a rigorous definition of
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A generic and executable formalization of signature-based Gröbner basis algorithms J. Symb. Comput. (IF 0.673) Pub Date : 2020-12-04 Alexander Maletzky
We present a generic and executable formalization of signature-based algorithms (such as Faugère's F5) for computing Gröbner bases, as well as their mathematical background, in the Isabelle/HOL proof assistant. Said algorithms are currently the best known algorithms for computing Gröbner bases in terms of computational efficiency. The formal development attempts to be as generic as possible, generalizing
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On Bézout inequalities for non-homogeneous polynomial ideals J. Symb. Comput. (IF 0.673) Pub Date : 2020-11-25 Amir Hashemi, Joos Heintz, Luis M. Pardo, Pablo Solernó
We introduce a “workable” notion of degree for non-homogeneous polynomial ideals and formulate and prove ideal theoretic Bézout inequalities for the sum of two ideals in terms of this notion of degree and the degree of generators.
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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
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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 initial 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
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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
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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
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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
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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
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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(degf
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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 et al. (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 algorithms
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Wasserstein distance to independence models J. Symb. Comput. (IF 0.673) Pub Date : 2020-10-23 Türkü Özlüm Çelik, Asgar Jamneshan, Guido Montúfar, Bernd Sturmfels, Lorenzo Venturello
An independence model for discrete random variables is a Segre-Veronese variety in a probability simplex. Any metric on the set of joint states of the random variables induces a Wasserstein metric on the probability simplex. The unit ball of this polyhedral norm is dual to the Lipschitz polytope. Given any data distribution, we seek to minimize its Wasserstein distance to a fixed independence model
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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, 2013).
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New ways to multiply 3 × 3-matrices J. Symb. Comput. (IF 0.673) Pub Date : 2020-10-20 Marijn J.H. Heule, Manuel Kauers, Martina Seidl
It is known since the 1970s that no more than 23 multiplications are required for computing the product of two 3×3-matrices. For non-commutative coefficient rings, it is not known whether it can also be done with fewer multiplications. However, there are several mutually inequivalent ways of doing the job with 23 multiplications. In this article, we extend this list considerably by providing more than
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Quasi-independence models with rational maximum likelihood estimator J. Symb. Comput. (IF 0.673) Pub Date : 2020-11-01 Jane Ivy Coons, Seth Sullivant
We classify the two-way quasi-independence models (independence models with structural zeros) that have rational maximum likelihood estimators, or MLEs. We give a necessary and sufficient condition on the bipartite graph associated to the model for the MLE to be rational. In this case, we give an explicit formula for the MLE in terms of combinatorial features of this graph. We also use the Horn uniformization
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Exact algorithms for semidefinite programs with degenerate feasible set J. Symb. Comput. (IF 0.673) Pub Date : 2020-11-19 Didier Henrion, Simone Naldi, Mohab Safey El Din
Given symmetric matrices A0,A1,…,An of size m with rational entries, the set of real vectors x=(x1,…,xn) such that the matrix A0+x1A1+⋯+xnAn has non-negative eigenvalues is called a spectrahedron. Minimization of linear functions over spectrahedra is called semidefinite programming. Such problems appear frequently in control theory and real algebra, especially in the context of nonnegativity certificates
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Analytic integrability of quasi-homogeneous systems via the Yoshida method J. Symb. Comput. (IF 0.673) Pub Date : 2020-11-23 Belén García, Jaume Llibre, Antón Lombardero, Jesús S. Pérez del Río
The objective of this paper is twofold. First we do a survey on what we call the Yoshida method for studying the analytic first integrals of the quasi-homogeneous polynomial differential systems. After, we apply the Yoshida method for studying the analytic first integrals of all the quasi-homogeneous polynomial differential systems in R3 of degree 2.
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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
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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
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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
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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
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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
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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
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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 (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 for the presentation
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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
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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
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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
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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
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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
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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.
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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
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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
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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.
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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