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The integral closure of a primary ideal is not always primary J. Symb. Comput. (IF 0.7) Pub Date : 2024-03-06 Nan Li, Zijia Li, Zhi-Hong Yang, Lihong Zhi
In , Krull asked if the integral closure of a primary ideal is still primary. Fifty years later, Huneke partially answered this question by giving a primary polynomial ideal whose integral closure is not primary in a regular local ring of characteristic . We provide counterexamples to Krull's question regarding polynomial rings over any fields. We also find that the Jacobian ideal of the polynomial
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Solving polynomial systems over non-fields and applications to modular polynomial factoring J. Symb. Comput. (IF 0.7) Pub Date : 2024-03-05 Sayak Chakrabarti, Ashish Dwivedi, Nitin Saxena
We study the problem of solving a system of polynomials in variables over the ring of integers modulo a prime-power . The problem over finite fields is well studied in varied parameter settings. For small characteristic , Lokshtanov et al. (SODA'17) initiated the study, for degree systems, to improve the exhaustive search complexity of to ; which currently is improved to in Dinur (SODA'21). For large
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A computational approach to almost-inner derivations J. Symb. Comput. (IF 0.7) Pub Date : 2024-03-04 Heiko Dietrich, Willem A. de Graaf
We present a computational approach to determine the space of almost-inner derivations of a finite dimensional Lie algebra given by a structure constant table. We also present an example of a Lie algebra for which the quotient algebra of the almost-inner derivations modulo the inner derivations is non-abelian. This answers a question of Kunyavskii and Ostapenko.
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Stabilized recovery and model reduction for multivariate exponential polynomials J. Symb. Comput. (IF 0.7) Pub Date : 2024-03-01 Juan Manuel Peña, Tomas Sauer
Recovery of multivariate exponential polynomials, i.e., the multivariate version of Prony's problem, can be stabilized by using more than the minimally needed multiinteger samples of the function. We present an algorithm that takes into account this extra information and prove a backward error estimate for the algebraic recovery method SMILE. In addition, we give a method to approximate data by an
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Computing a group action from the class field theory of imaginary hyperelliptic function fields J. Symb. Comput. (IF 0.7) Pub Date : 2024-03-01 Antoine Leudière, Pierre-Jean Spaenlehauer
We explore algorithmic aspects of a simply transitive commutative group action coming from the class field theory of imaginary hyperelliptic function fields. Namely, the Jacobian of an imaginary hyperelliptic curve defined over acts on a subset of isomorphism classes of Drinfeld modules. We describe an algorithm to compute the group action efficiently. This is a function field analog of the Couvei
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Matrix factorizations of the discriminant of S J. Symb. Comput. (IF 0.7) Pub Date : 2024-02-23 Eleonore Faber, Colin Ingalls, Simon May, Marco Talarico
Consider the symmetric group acting as a reflection group on the polynomial ring where is a field, such that Char() does not divide !. We use Higher Specht polynomials to construct matrix factorizations of the discriminant of this group action: these matrix factorizations are indexed by partitions of and respect the decomposition of the coinvariant algebra into isotypical components. The maximal Cohen–Macaulay
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An improved complexity bound for computing the topology of a real algebraic space curve J. Symb. Comput. (IF 0.7) Pub Date : 2024-02-21 Jin-San Cheng, Kai Jin, Marc Pouget, Junyi Wen, Bingwei Zhang
We propose a new algorithm to compute the topology of a real algebraic space curve. The novelties of this algorithm are a new technique to achieve the lifting step which recovers points of the space curve in each plane fiber from several projections and a weaker notion of generic position. As distinct to previous work, our does not require that -critical points have different -coordinates. The complexity
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Rational solutions to the first order difference equations in the bivariate difference field J. Symb. Comput. (IF 0.7) Pub Date : 2024-02-09 Qing-Hu Hou, Yarong Wei
Inspired by Karr's algorithm, we consider the summations involving a sequence satisfying a recurrence of order two. The structure of such summations provides an algebraic framework for solving the difference equations of form in the bivariate difference field , where are known binary functions of , , and , are two algebraically independent transcendental elements, is a transformation that satisfies
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Tensor decompositions on simplicial complexes with invariance J. Symb. Comput. (IF 0.7) Pub Date : 2024-01-19 Gemma De las Cuevas, Matt Hoogsteder Riera, Tim Netzer
Tensors are ubiquitous in mathematics and the sciences, as they allow to store information in a concise way. Decompositions of tensors may give insights into their structure and complexity. In this work, we develop a new framework for decompositions of tensors, taking into account invariance, positivity and a geometric arrangement of their local spaces. We define an invariant decomposition with indices
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Computing the binomial part of a polynomial ideal J. Symb. Comput. (IF 0.7) Pub Date : 2024-01-14 Martin Kreuzer, Florian Walsh
Given an ideal in a polynomial ring over a field , we present a complete algorithm to compute the binomial part of , i.e., the subideal of generated by all monomials and binomials in . This is achieved step-by-step. First we collect and extend several algorithms for computing exponent lattices in different kinds of fields. Then we generalize them to compute exponent lattices of units in 0-dimensional
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A post-quantum key exchange protocol from the intersection of conics J. Symb. Comput. (IF 0.7) Pub Date : 2024-01-05 Alberto Alzati, Daniele Di Tullio, Manoj Gyawali, Alfonso Tortora
In this paper we present a key exchange protocol in which Alice and Bob have secret keys given by two conics embedded in a large ambient space by means of the Veronese embedding and public keys given by hyperplanes containing the embedded curves. Both of them construct some common invariants given by the intersection of two conics.
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Theta nullvalues of supersingular Abelian varieties J. Symb. Comput. (IF 0.7) Pub Date : 2023-12-29 Andreas Pieper
Let η be a polarization with connected kernel on a superspecial abelian variety Eg. We give a sufficient criterion which allows the computation of the theta nullvalues of any quotient of Eg by a maximal isotropic subgroup scheme of ker(η) effectively. This criterion is satisfied in many situations studied by Li and Oort (1998). We used our method to implement an algorithm that computes supersingular
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A note on the relation between recognisable series and regular sequences, and their minimal linear representations J. Symb. Comput. (IF 0.7) Pub Date : 2023-12-28 Clemens Heuberger, Daniel Krenn, Gabriel F. Lipnik
In this note, we precisely elaborate the connection between recognisable series (in the sense of Berstel and Reutenauer) and q-regular sequences (in the sense of Allouche and Shallit) via their linear representations. In particular, we show that the minimisation algorithm for recognisable series can also be used to minimise linear representations of q-regular sequences.
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Axioms for a theory of signature bases J. Symb. Comput. (IF 0.7) Pub Date : 2023-12-18 Pierre Lairez
Twenty years after the discovery of the F5 algorithm, Gröbner bases with signatures are still challenging to understand and to adapt to different settings. This contrasts with Buchberger's algorithm, which we can bend in many directions keeping correctness and termination obvious. I propose an axiomatic approach to Gröbner bases with signatures with the purpose of uncoupling the theory and the algorithms
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Computing primitive idempotents in finite commutative rings and applications J. Symb. Comput. (IF 0.7) Pub Date : 2023-12-15 Mugurel Barcau, Vicenţiu Paşol
In this paper, we compute an algebraic decomposition of black-box rings in the generic ring model. More precisely, we explicitly decompose a black-box ring as a direct product of a nilpotent black-box ring and unital local black-box rings, by computing all its primitive idempotents. The algorithm presented in this paper uses quantum subroutines for the computation of the p-power parts of a black-box
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Levelwise construction of a single cylindrical algebraic cell J. Symb. Comput. (IF 0.7) Pub Date : 2023-12-05 Jasper Nalbach, Erika Ábrahám, Philippe Specht, Christopher W. Brown, James H. Davenport, Matthew England
Satisfiability modulo theories (SMT) solvers check the satisfiability of quantifier-free first-order logic formulae over different theories. We consider the theory of non-linear real arithmetic where the formulae are logical combinations of polynomial constraints. Here a commonly used tool is the cylindrical algebraic decomposition (CAD) to decompose the real space into cells where the constraints
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Fast computation of the centralizer of a permutation group in the symmetric group J. Symb. Comput. (IF 0.7) Pub Date : 2023-12-05 Rok Požar
Let G be a permutation group acting on a set Ω. Best known algorithms for computing the centralizer of G in the symmetric group on Ω are all based on the same general approach that involves solving the following two fundamental problems: given a G-orbit Δ of size n, compute the centralizer of the restriction of G to Δ in the symmetric group on Δ; and given two G-orbits Δ and Δ′ each of size n, find
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Two-step Newton's method for deflation-one singular zeros of analytic systems J. Symb. Comput. (IF 0.7) Pub Date : 2023-11-20 Kisun Lee, Nan Li, Lihong Zhi
We propose a two-step Newton's method for refining an approximation of a singular zero whose deflation process terminates after one step, also known as a deflation-one singularity. Given an isolated singular zero of a square analytic system, our algorithm exploits an invertible linear operator obtained by combining the Jacobian and a projection of the Hessian in the direction of the kernel of the Jacobian
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Toward finiteness of central configurations for the planar six-body problem by symbolic computations. (I) Determine diagrams and orders J. Symb. Comput. (IF 0.7) Pub Date : 2023-11-17 Ke-Ming Chang, Kuo-Chang Chen
In a series of papers we develop symbolic computation algorithms to investigate finiteness of central configurations for the planar n-body problem. Our approach is based on Albouy-Kaloshin's work on finiteness of central configurations for the 5-body problems. In their paper, bicolored graphs called zw-diagrams were introduced for possible scenarios when the finiteness conjecture fails, and proving
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Explainable AI Insights for Symbolic Computation: A case study on selecting the variable ordering for cylindrical algebraic decomposition J. Symb. Comput. (IF 0.7) Pub Date : 2023-11-15 Lynn Pickering, Tereso del Río Almajano, Matthew England, Kelly Cohen
In recent years there has been increased use of machine learning (ML) techniques within mathematics, including symbolic computation where it may be applied safely to optimise or select algorithms. This paper explores whether using explainable AI (XAI) techniques on such ML models can offer new insight for symbolic computation, inspiring new implementations within computer algebra systems that do not
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Squarefree normal representation of zeros of zero-dimensional polynomial systems J. Symb. Comput. (IF 0.7) Pub Date : 2023-11-14 Juan Xu, Dongming Wang, Dong Lu
For any zero-dimensional polynomial ideal I and any nonzero polynomial F, this paper shows that the union of the multi-set of zeros of the ideal sum I+〈F〉 and that of the ideal quotient I:〈F〉 is equal to the multi-set of zeros of I, where zeros are counted with multiplicities. Based on this zero relation and the computation of Gröbner bases, a complete multiplicity-preserved algorithm is proposed to
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Syzygies, constant rank, and beyond J. Symb. Comput. (IF 0.7) Pub Date : 2023-11-14 Marc Härkönen, Lisa Nicklasson, Bogdan Raiţă
We study linear PDE with constant coefficients. The constant rank condition on a system of linear PDEs with constant coefficients is often used in the theory of compensated compactness. While this is a purely linear algebraic condition, the nonlinear algebra concept of primary decomposition is another important tool for studying such system of PDEs. In this paper we investigate the connection between
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Representation of non-special curves of genus 5 as plane sextic curves and its application to finding curves with many rational points J. Symb. Comput. (IF 0.7) Pub Date : 2023-09-25 Momonari Kudo, Shushi Harashita
In algebraic geometry, it is important to provide effective parametrizations for families of curves, both in theory and in practice. In this paper, we present such an effective parametrization for the moduli of genus-5 curves that are neither hyperelliptic nor trigonal. Subsequently, we construct an algorithm for a complete enumeration of non-special genus-5 curves having more rational points than
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Formations of finite groups in polynomial time: F-residuals and F-subnormality J. Symb. Comput. (IF 0.7) Pub Date : 2023-09-25 Viachaslau I. Murashka
For a wide family of formations F it is proved that the F-residual of a permutation finite group can be computed in polynomial time. Moreover, if in the previous case F is hereditary, then the F-subnormality of a subgroup can be checked in polynomial time.
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MacWilliams' Extension Theorem for rank-metric codes J. Symb. Comput. (IF 0.7) Pub Date : 2023-09-01 Elisa Gorla, Flavio Salizzoni
The MacWilliams' Extension Theorem is a classical result by Florence Jessie MacWilliams. It shows that every linear isometry between linear block-codes endowed with the Hamming distance can be extended to a linear isometry of the ambient space. Such an extension fails to exist in general for rank-metric codes, that is, one can easily find examples of linear isometries between rank-metric codes which
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Segre-driven radicality testing J. Symb. Comput. (IF 0.7) Pub Date : 2023-08-23 Martin Helmer, Elias Tsigaridas
We present a probabilistic algorithm to test if a homogeneous polynomial ideal I defining a scheme X in Pn is radical using Segre classes and other geometric notions from intersection theory which is applicable for certain classes of ideals. If all isolated primary components of the scheme X are reduced and it has no embedded components outside of the singular locus of Xred=V(I), then the algorithm
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Invariants of SDP exactness in quadratic programming J. Symb. Comput. (IF 0.7) Pub Date : 2023-08-22 Julia Lindberg, Jose Israel Rodriguez
In this paper we study the Shor relaxation of quadratic programs by fixing a feasible set and considering the space of objective functions for which the Shor relaxation is exact. We first give conditions under which this region is invariant under the choice of generators defining the feasible set. We then describe this region when the feasible set is invariant under the action of a subgroup of the
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Free resolutions and Lefschetz properties of some Artin Gorenstein rings of codimension four J. Symb. Comput. (IF 0.7) Pub Date : 2023-08-21 Nancy Abdallah, Hal Schenck
In (Stanley, 1978), Stanley constructs an example of an Artinian Gorenstein (AG) ring A with non-unimodal H-vector (1,13,12,13,1). Migliore-Zanello show in (Migliore and Zanello, 2017) that for regularity r=4, Stanley's example has the smallest possible codimension c for an AG ring with non-unimodal H-vector. The weak Lefschetz property (WLP) has been much studied for AG rings; it is easy to show that
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Algebraic number fields and the LLL algorithm J. Symb. Comput. (IF 0.7) Pub Date : 2023-08-18 M.J. Uray
In this paper we analyze the computational costs of various operations and algorithms in algebraic number fields using exact arithmetic. Let K be an algebraic number field. In the first half of the paper, we calculate the running time and the size of the output of many operations in K in terms of the size of the input and the parameters of K. We include some earlier results about these, but we go further
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The critical curvature degree of an algebraic variety J. Symb. Comput. (IF 0.7) Pub Date : 2023-08-16 Emil Horobeţ
In this article we study the complexity involved in the computation of the reach in arbitrary dimension and in particular the computation of the critical spherical curvature points of an arbitrary algebraic variety. We present properties of the critical spherical curvature points as well as an algorithm for computing them.
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Universal equations for maximal isotropic Grassmannians J. Symb. Comput. (IF 0.7) Pub Date : 2023-08-16 Tim Seynnaeve, Nafie Tairi
The isotropic Grassmannian parametrizes isotropic subspaces of a vector space equipped with a quadratic form. In this paper, we show that any maximal isotropic Grassmannian in its Plücker embedding can be defined by pulling back the equations of Griso(3,7) or Griso(4,8).
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A tree-based algorithm for the integration of monomials in the Chow ring of the moduli space of stable marked curves of genus zero J. Symb. Comput. (IF 0.7) Pub Date : 2023-08-11 Jiayue Qi
The Chow ring of the moduli space of marked rational curves is generated by Keel's divisor classes. The top graded part of this Chow ring is isomorphic to the integers, generated by the class of a single point. In this paper, we give an equivalent graphical characterization on the monomials in this Chow ring, as well as the characterization on the algebraic reduction on such monomials. Moreover, we
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Nash conditional independence curve J. Symb. Comput. (IF 0.7) Pub Date : 2023-08-01 Irem Portakal, Javier Sendra–Arranz
We study the Spohn conditional independence (CI) variety CX of an n-player game X for undirected graphical models on n binary random variables consisting of one edge. For a generic game, we show that CX is a smooth irreducible complete intersection curve (Nash conditional independence curve) in the Segre variety (P1)n−2×P3 and we give an explicit formula for its degree and genus. We prove two universality
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Logarithmic Voronoi cells for Gaussian models J. Symb. Comput. (IF 0.7) Pub Date : 2023-08-01 Yulia Alexandr, Serkan Hoşten
We extend the theory of logarithmic Voronoi cells to Gaussian statistical models. In general, a logarithmic Voronoi cell at a point on a Gaussian model is a convex set contained in its log-normal spectrahedron. We show that for models of ML degree one and linear covariance models the two sets coincide. In particular, they are equal for both directed and undirected graphical models. We introduce decomposition
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Rational dual certificates for weighted sums-of-squares polynomials with boundable bit size J. Symb. Comput. (IF 0.7) Pub Date : 2023-07-31 Maria M. Davis, Dávid Papp
In Davis and Papp (2022), the authors introduced the concept of dual certificates of (weighted) sum-of-squares polynomials, which are vectors from the dual cone of weighted sums of squares (WSOS) polynomials that can be interpreted as nonnegativity certificates. This initial theoretical work showed that for every polynomial in the interior of a WSOS cone, there exists a rational dual certificate proving
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Isolating all the real roots of a mixed trigonometric-polynomial J. Symb. Comput. (IF 0.7) Pub Date : 2023-07-20 Rizeng Chen, Haokun Li, Bican Xia, Tianqi Zhao, Tao Zheng
Mixed trigonometric-polynomials (MTPs) are functions of the form f(x,sinx,cosx) where f is a trivariate polynomial with rational coefficients, and the argument x ranges over the reals. In this paper, an algorithm “isolating” all the real roots of an MTP is provided and implemented. It automatically divides the real roots into two parts: one consists of finitely many roots in an interval [μ−,μ+] while
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Deformations of half-canonical Gorenstein curves in codimension four J. Symb. Comput. (IF 0.7) Pub Date : 2023-07-20 Patience Ablett, Stephen Coughlan
Recent work of Ablett (2021) and Kapustka, Kapustka, Ranestad, Schenck, Stillman and Yuan (2021) outlines a number of constructions for singular Gorenstein codimension four varieties. Earlier work of Coughlan, Gołȩbiowski, Kapustka and Kapustka (2016) details a series of nonsingular Gorenstein codimension four constructions with different Betti tables. In this paper we exhibit a number of flat deformations
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On the computation of rational solutions of linear integro-differential equations with polynomial coefficients J. Symb. Comput. (IF 0.7) Pub Date : 2023-07-20 Moulay Barkatou, Thomas Cluzeau
We develop the first algorithm for computing rational solutions of scalar integro-differential equations with polynomial coefficients. It starts by finding the possible poles of a rational solution. Then, bounding the order of each pole and solving an algebraic linear system, we compute the singular part of rational solutions at each possible pole. Finally, using partial fraction decomposition, the
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Algebraic optimization of sequential decision problems J. Symb. Comput. (IF 0.7) Pub Date : 2023-07-11 Mareike Dressler, Marina Garrote-López, Guido Montúfar, Johannes Müller, Kemal Rose
We study the optimization of the expected long-term reward in finite partially observable Markov decision processes over the set of stationary stochastic policies. In the case of deterministic observations, also known as state aggregation, the problem is equivalent to optimizing a linear objective subject to quadratic constraints. We characterize the feasible set of this problem as the intersection
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Skew-polynomial-sparse matrix multiplication J. Symb. Comput. (IF 0.7) Pub Date : 2023-07-03 Qiao-Long Huang, Ke Ye, Xiao-Shan Gao
Based on the observation that Q(p−1)×(p−1) is isomorphic to a quotient skew polynomial ring, we propose a new deterministic algorithm for (p−1)×(p−1) matrix multiplication over Q, where p is a prime number. The algorithm has complexity O(Tω−2p2), where T≤p−1 is a parameter determined by the skew-polynomial-sparsity of input matrices and ω is the asymptotic exponent of matrix multiplication. Here a
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Hirota varieties and rational nodal curves J. Symb. Comput. (IF 0.7) Pub Date : 2023-06-28 Claudia Fevola, Yelena Mandelshtam
The Hirota variety parameterizes solutions to the KP equation arising from a degenerate Riemann theta function. In this work, we study in detail the Hirota variety arising from a rational nodal curve. Of particular interest is the irreducible subvariety defined as the image of a parameterization map, we call this the main component. Proving that this is an irreducible component of the Hirota variety
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An effective decomposition theorem for Schubert varieties J. Symb. Comput. (IF 0.7) Pub Date : 2023-06-28 Francesca Cioffi, Davide Franco, Carmine Sessa
Given a Schubert variety S contained in a Grassmannian Gk(Cl), we show how to obtain further information on the direct summands of the derived pushforward Rπ⁎QS˜ given by the application of the decomposition theorem to a suitable resolution of singularities π:S˜→S. As a by-product, Poincaré polynomial expressions are obtained along with an algorithm which computes the unknown terms in such expressions
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Sagbi combinatorics of maximal minors and a Sagbi algorithm J. Symb. Comput. (IF 0.7) Pub Date : 2023-06-28 Winfried Bruns, Aldo Conca
The maximal minors of a matrix of indeterminates are a universal Gröbner basis by a theorem of Bernstein, Sturmfels and Zelevinsky. On the other hand it is known that they are not always a universal Sagbi basis. By an experimental approach we discuss their behavior under varying monomial orders and their extensions to Sagbi bases. These experiments motivated a new implementation of the Sagbi algorithm
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Sum-of-squares certificates for Vizing's conjecture via determining Gröbner bases J. Symb. Comput. (IF 0.7) Pub Date : 2023-06-19 Elisabeth Gaar, Melanie Siebenhofer
The famous open Vizing conjecture claims that the domination number of the Cartesian product graph of two graphs G and H is at least the product of the domination numbers of G and H. Recently Gaar, Krenn, Margulies and Wiegele used the graph class G of all graphs with nG vertices and domination number kG and reformulated Vizing's conjecture as the problem that for all graph classes G and H the Vizing
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Computing with Tarski formulas and semi-algebraic sets in a web browser J. Symb. Comput. (IF 0.7) Pub Date : 2023-06-16 Zoltán Kovács, Christopher Brown, Tomás Recio, Róbert Vajda
We report on successfully porting the Tarski system for computing with Tarski formulas (Boolean combinations of polynomial sign conditions over the real numbers) to Javascript, thereby allowing it to run inside a browser. Tarski is an open-source software package, written in C/C++, that provides operations like formula simplification and quantifier elimination for Tarski formulas. Leveraging the Emscripten
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Computing roadmaps in unbounded smooth real algebraic sets I: Connectivity results J. Symb. Comput. (IF 0.7) Pub Date : 2023-06-08 Rémi Prébet, Mohab Safey El Din, Éric Schost
Answering connectivity queries in real algebraic sets is a fundamental problem in effective real algebraic geometry that finds many applications in e.g. robotics where motion planning issues are topical. This computational problem is tackled through the computation of so-called roadmaps which are real algebraic subsets of the set V under study, of dimension at most one, and which have a connected intersection
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The Smith normal form and reduction of weakly linear matrices J. Symb. Comput. (IF 0.7) Pub Date : 2023-06-08 Jinwang Liu, Dongmei Li, Tao Wu
The reduction of a multidimensional system is closely related to the reduction of a multivariate polynomial matrix, for which the Smith normal form of the matrix plays a key role. In this paper, we investigate the reduction of weakly linear multivariate polynomial matrices to their Smith normal forms. Using hierarchical-recursive method and Quillen-Suslin Theorem, we derive some necessary and sufficient
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A “pseudo-polynomial” algorithm for the Frobenius number and Gröbner basis J. Symb. Comput. (IF 0.7) Pub Date : 2023-06-08 Marcel Morales, Nguyen Thi Dung
Given n⩾2 and a1,…,an∈N. Let S=〈a1,…,an〉 be a semigroup. The aim of this paper is to give an effective pseudo-polynomial algorithm on a1, which computes the Apéry set and the Frobenius number of S. We also find the Gröbner basis of the toric ideal defined by S, for the weighted degree reverse lexicographical order ≺w to x1,…,xn, without using Buchberger's algorithm. As an application we introduce and
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An algorithmic approach based on generating trees for enumerating pattern-avoiding inversion sequences J. Symb. Comput. (IF 0.7) Pub Date : 2023-05-18 Ilias Kotsireas, Toufik Mansour, Gökhan Yıldırım
We introduce an algorithmic approach based on a generating tree method for enumerating the inversion sequences with various pattern-avoidance restrictions. For a given set of patterns, we propose an algorithm that outputs either an accurate description of the succession rules of the corresponding generating tree or an ansatz. By using this approach, we determine the generating trees for the pattern
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The span of singular tuples of a tensor beyond the boundary format J. Symb. Comput. (IF 0.7) Pub Date : 2023-05-10 Luca Sodomaco, Ettore Teixeira Turatti
A singular k-tuple of a tensor T of format (n1,…,nk) is essentially a complex critical point of the distance function from T constrained to the cone of tensors of format (n1,…,nk) of rank at most one. A generic tensor has finitely many complex singular k-tuples, and their number depends only on the tensor format. Furthermore, if we fix the first k−1 dimensions ni, then the number of singular k-tuples
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Voronoi diagrams of algebraic varieties under polyhedral norms J. Symb. Comput. (IF 0.7) Pub Date : 2023-05-09 Adrian Becedas, Kathlén Kohn, Lorenzo Venturello
We study Voronoi diagrams of manifolds and varieties with respect to polyhedral norms. We provide upper and lower bounds on the dimensions of Voronoi cells. For algebraic varieties, we count their full-dimensional Voronoi cells. As an application, we consider the polyhedral Wasserstein distance between discrete probability distributions.
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Sextactic points on the Fermat cubic curve and arrangements of conics J. Symb. Comput. (IF 0.7) Pub Date : 2023-04-28 Tomasz Szemberg, Justyna Szpond
The purpose of this note is to report, in narrative rather than rigorous style, about the nice geometry of 6-division points on the Fermat cubic F and various conics naturally attached to them. Most facts presented here were derived by symbolic algebra programs and the idea of the note is to propose a research direction in the search of conceptual proofs of facts stated here and their generalisations
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On cyclic codes over Zq[u]/〈u2〉 and their enumeration J. Symb. Comput. (IF 0.7) Pub Date : 2023-04-24 Fatih Temiz, Irfan Siap
In this study, we determine the structure of cyclic codes over the ring Zq[u]/〈u2〉 which is isomorphic to R=Zq+uZq where q=ps, p is a prime, s is a positive integer, and u2=0. This is equivalent to determining the algebraic structure of ideals of the polynomial quotient ring R[x]/〈xn−1〉, which is addressed in this paper completely. By establishing the structure of ideals of R[x]/〈xn−1〉 with gcd(p
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Critical configurations for two projective views, a new approach J. Symb. Comput. (IF 0.7) Pub Date : 2023-04-18 Martin Bråtelund
This article develops new techniques to classify critical configurations for 3D scene reconstruction from images taken by unknown cameras. Generally, all information can be uniquely recovered if enough images and image points are provided, but there are certain cases where unique recovery is impossible; these are called critical configurations. In this paper, we use an algebraic approach to study the
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Massively parallel computation of tropical varieties, their positive part, and tropical Grassmannians J. Symb. Comput. (IF 0.7) Pub Date : 2023-04-13 Dominik Bendle, Janko Böhm, Yue Ren, Benjamin Schröter
We present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of symmetries using the workflow management system GPI-Space and the computer algebra system Singular. We determine the tropical Grassmannian TGr0(3,8). Our implementation works efficiently on up to 840 cores, computing the 14763 orbits of maximal cones under the canonical S8-action in
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Computing tropical bitangents to smooth quartic curves in polymake J. Symb. Comput. (IF 0.7) Pub Date : 2023-04-13 Alheydis Geiger, Marta Panizzut
In this article we introduce the recently developed polymake extension TropicalQuarticCurves and its associated database entry in polyDB dealing with smooth tropical quartic curves. We report on algorithms implemented to analyze tropical bitangents and their lifting conditions over real closed valued fields. The new functions and data were used by the authors to provide a tropical proof of Plücker
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A counterexample to a conjecture on simultaneous Waring identifiability J. Symb. Comput. (IF 0.7) Pub Date : 2023-04-11 Elena Angelini
The new identifiable case appeared in Angelini et al. (2018), together with the analysis on simultaneous identifiability of pairs of ternary forms recently developed in Beorchia and Galuppi (2022), suggested the following conjecture towards a complete classification of all simultaneous Waring identifiable cases: for any d≥2, the general polynomial vectors consisting of d−1 ternary forms of degree d
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Toric geometry of entropic regularization J. Symb. Comput. (IF 0.7) Pub Date : 2023-04-07 Bernd Sturmfels, Simon Telen, François-Xavier Vialard, Max von Renesse
Entropic regularization is a method for large-scale linear programming. Geometrically, one traces intersections of the feasible polytope with scaled toric varieties, starting at the Birch point. We compare this to log-barrier methods, with reciprocal linear spaces, starting at the analytic center. We revisit entropic regularization for unbalanced optimal transport, and we develop the use of optimal
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Counting solutions of a polynomial system locally and exactly J. Symb. Comput. (IF 0.7) Pub Date : 2023-04-07 Ruben Becker, Michael Sagraloff
In this paper, we propose a symbolic-numeric algorithm to count the number of solutions of a zero-dimensional square polynomial system within a local region. We show that the algorithm succeeds under the condition that the region is sufficiently small and well-isolating for a k-fold solution z of the system. In our analysis, we derive a bound on the size of the region that guarantees success. We further