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  • Resultants over principal Artinian rings
    arXiv.cs.SC Pub Date : 2020-04-07
    Claus Fieker; Tommy Hofmann; Carlo Sircana

    The resultant of two univariate polynomials is an invariant of great importance in commutative algebra and vastly used in computer algebra systems. Here we present an algorithm to compute it over Artinian principal rings with a modified version of the Euclidean algorithm. Using the same strategy, we show how the reduced resultant and a pair of B\'ezout coefficient can be computed. Particular attention

  • Neural Analogical Matching
    arXiv.cs.SC Pub Date : 2020-04-07
    Maxwell Crouse; Constantine Nakos; Ibrahim Abdelaziz; Kenneth Forbus

    Analogy is core to human cognition. It allows us to solve problems based on prior experience, it governs the way we conceptualize new information, and it even influences our visual perception. The importance of analogy to humans has made it an active area of research in the broader field of artificial intelligence, resulting in data-efficient models that learn and reason in human-like ways. While analogy

  • Interpolation of Dense and Sparse Rational Functions and other Improvements in $\texttt{FireFly}$
    arXiv.cs.SC Pub Date : 2020-04-03
    Jonas Klappert; Sven Yannick Klein; Fabian Lange

    We present the main improvements and new features in version $\texttt{2.0}$ of the open-source $\texttt{C++}$ library $\texttt{FireFly}$ for the interpolation of rational functions. This includes algorithmic improvements, e.g. a hybrid algorithm for dense and sparse rational functions and an algorithm to identify and remove univariate factors. The new version is applied to a Feynman-integral reduction

  • Stream/block ciphers, difference equations and algebraic attacks
    arXiv.cs.SC Pub Date : 2020-03-28
    Roberto La Scala; Sharwan K. Tiwari

    In this paper we introduce a general class of stream and block ciphers that are defined by means of systems of (ordinary) explicit difference equations over a finite field. We call this class "difference ciphers". Many important ciphers such as systems of LFSRs, Trivium/Bivium and Keeloq are difference ciphers. To the purpose of studying their underlying explicit difference systems, we introduce key

  • Fast Encoding of AG Codes over $C_{ab}$ Curves
    arXiv.cs.SC Pub Date : 2020-03-30
    Peter Beelen; Johan Rosenkilde; Grigory Solomatov

    We investigate algorithms for encoding of one-point algebraic geometry (AG) codes over certain plane curves called $C_{ab}$ curves, as well as algorithms for inverting the encoding map, which we call "unencoding". Some $C_{ab}$ curves have many points or are even maximal, e.g. the Hermitian curve. Our encoding resp. unencoding algorithms have complexity $\tilde{O}(n^{3/2})$ resp. $\tilde{O}(qn)$ for

  • Generic bivariate multi-point evaluation, interpolation and modular composition with precomputation
    arXiv.cs.SC Pub Date : 2020-03-27
    Vincent Neiger; Johan Rosenkilde; Grigory Solomatov

    If $\mathbb{K}$ is a large enough field and $\mathcal{P} \subset \mathbb{K}^2$ is a fixed, generic set of points, which is available for precomputation, we show how to compute all the evaluations of any dense polynomial $f$ on $\mathcal{P}$ in quasi-linear time. Similarly, in quasi-linear time then given interpolation constraints on $\mathcal{P}$ and a target $y$-degree, we compute an $f$ having those

  • A Nonexistence Certificate for Projective Planes of Order Ten with Weight 15 Codewords
    arXiv.cs.SC Pub Date : 2019-11-11
    Curtis Bright; Kevin Cheung; Brett Stevens; Dominique Roy; Ilias Kotsireas; Vijay Ganesh

    Using techniques from the fields of symbolic computation and satisfiability checking we verify one of the cases used in the landmark result that projective planes of order ten do not exist. In particular, we show that there exist no projective planes of order ten that generate codewords of weight fifteen, a result first shown in 1973 via an exhaustive computer search. We provide a simple satisfiability

  • Moment State Dynamical Systems for Nonlinear Chance-Constrained Motion Planning
    arXiv.cs.SC Pub Date : 2020-03-23
    Allen Wang; Ashkan Jasour; Brian Williams

    Chance-constrained motion planning requires uncertainty in dynamics to be propagated into uncertainty in state. When nonlinear models are used, Gaussian assumptions on the state distribution do not necessarily apply since almost all random variables propagated through nonlinear dynamics results in non-Gaussian state distributions. To address this, recent works have developed moment-based approaches

  • On Exact Reznick, Hilbert-Artin and Putinar's Representations
    arXiv.cs.SC Pub Date : 2018-11-25
    Victor Magron; Mohab Safey El Din

    We consider the problem of finding 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 for polynomials lying in the interior of the SOS cone. It computes an approximate SOS decomposition for a perturbation

  • An Algorithm for Computing a Minimal Comprehensive Gröbner\, Basis of a Parametric Polynomial System
    arXiv.cs.SC Pub Date : 2020-03-17
    Deepak Kapur; Yiming Yang

    An algorithm to generate a minimal comprehensive Gr\"obner\, basis of a parametric polynomial system from an arbitrary faithful comprehensive Gr\"obner\, system is presented. A basis of a parametric polynomial ideal is a comprehensive Gr\"obner\, basis if and only if for every specialization of parameters in a given field, the specialization of the basis is a Gr\"obner\, basis of the associated specialized

  • FunGrim: a symbolic library for special functions
    arXiv.cs.SC Pub Date : 2020-03-13
    Fredrik JohanssonLFANT

    We present the Mathematical Functions Grimoire (FunGrim), a website and database of formulas and theorems for special functions. We also discuss the symbolic computation library used as the backend and main development tool for FunGrim, and the Grim formula language used in these projects to represent mathematical content semantically.

  • Experimental Evaluation of a Method to Simplify Expressions
    arXiv.cs.SC Pub Date : 2020-03-13
    Baudouin Le Charlier

    We present a method to simplify expressions in the context of an equational theory. The basic ideas and concepts of the method have been presented previously elsewhere but here we tackle the difficult task of making it efficient in practice, in spite of its great generality. We first recall the notion of a collection of structures, which allows us to manipulate very large (possibly infinite) sets of

  • Transforming ODEs and PDEs with radical coefficients into rational coefficients
    arXiv.cs.SC Pub Date : 2020-03-13
    Jorge Caravantes; J. Rafael Sendra; David Sevilla; Carlos Villarino

    We present an algorithm that transforms, if possible, a given ODE or PDE with radical function coefficients into one with rational coefficients by means of a rational change of variables. It also applies to systems of linear ODEs. It is based on previous work on reparametrization of radical algebraic varieties.

  • Algorithm to enumerate superspecial Howe curves of genus $4$
    arXiv.cs.SC Pub Date : 2020-03-09
    Momonari Kudo; Shushi Harashita

    A Howe curve is a curve of genus $4$ obtained as the fiber product over $\mathbf{P}^1$ of two elliptic curves. Any Howe curve is canonical. This paper provides an efficient algorithm to find superspecial Howe curves and that to enumerate their isomorphism classes. We discuss not only an algorithm to test the superspeciality but also an algorithm to test isomorphisms for Howe curves. Our algorithms

  • The Absent-Minded Passengers Problem: A Motivating Challenge Solved by Computer Algebra
    arXiv.cs.SC Pub Date : 2020-03-04
    Carsten Schneider

    In (S.B. Ekhad and D. Zeilberger, 2020) an exciting case study has been initiated in which experimental mathematics and symbolic computation are utilized to discover new properties concerning the so-called Absent-Minded Passengers Problem. Based on these results, Doron Zeilberger raised some challenging tasks to gain further probabilistic insight. In this note we report on this enterprise. In particular

  • Enhancing simultaneous rational function recovery: adaptive error correction capability and new bounds for applications
    arXiv.cs.SC Pub Date : 2020-03-03
    Eleonora Guerrini; Romain Lebreton; Ilaria Zappatore

    In this work we present some results that allow to improve the decoding radius in solving polynomial linear systems with errors in the scenario where errors are additive and randomly distributed over a finite field. The decoding radius depends on some bounds on the solution that we want to recover, so their overestimation could significantly decrease our error correction capability. For this reason

  • Effective Localization Using Double Ideal Quotient and Its Implementation
    arXiv.cs.SC Pub Date : 2020-02-29
    Yuki Ishihara; Kazuhiro Yokoyama

    In this paper, we propose a new method for localization of polynomial ideal, which we call "Local Primary Algorithm". For an ideal $I$ and a prime ideal $P$, our method computes a $P$-primary component of $I$ after checking if $P$ is associated with $I$ by using "double ideal quotient" $(I:(I:P))$ and its variants which give us a lot of information about localization of $I$.

  • A complexity chasm for solving sparse polynomial equations over $p$-adic fields
    arXiv.cs.SC Pub Date : 2020-02-29
    J. Maurice Rojas; Yuyu Zhu

    We reveal a complexity chasm, separating the trinomial and tetranomial cases, for solving univariate sparse polynomial equations over certain local fields. First, for any fixed field $K\in\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5,\ldots\}$, we prove that any polynomial $f\in\mathbb{Z}[x_1]$ with exactly $3$ monomial terms, degree $d$, and all coefficients having absolute value at most $H$, can be solved

  • Solving Satisfiability of Polynomial Formulas By Sample-Cell Projection
    arXiv.cs.SC Pub Date : 2020-03-01
    Haokun Li; Bican Xia

    A new algorithm for deciding the satisfiability of polynomial formulas over the real is proposed. The key point of the algorithm is a new projection operator, called sample-cell projection operator, custom-made for Conflict-Driven Clause Learning (CDCL)-style search. Although the new operator is also a CAD (Cylindrical Algebraic Decomposition)-like projection operator which computes the cell (not necessarily

  • Maximum Absolute Determinants of Upper Hessenberg Bohemian Matrices
    arXiv.cs.SC Pub Date : 2020-03-01
    Jonathan P. Keating; Ahmet Abdullah Keleş

    A matrix is called Bohemian if its entries are sampled from a finite set of integers. We determine the maximum absolute determinant of upper Hessenberg Bohemian Matrices for which the subdiagonal entries are fixed to be $1$ and upper triangular entries are sampled from $\{0,1,\cdots,n\}$, extending previous results for $n=1$ and $n=2$ and proving a recent conjecture of Fasi & Negri Porzio [8]. Furthermore

  • Modular Techniques for Effective Localization and Double Ideal Quotient
    arXiv.cs.SC Pub Date : 2020-03-01
    Yuki Ishihara

    By double ideal quotient, we mean $(I:(I:J))$ where ideals $I$ and $J$. In our previous work [11], double ideal quotient and its variants are shown to be very useful for checking prime divisor and generating primary component. Combining those properties, we can compute "direct localization" effectively, comparing with full primary decomposition. In this paper, we apply modular techniques effectively

  • Criteria for the numerical constant recognition
    arXiv.cs.SC Pub Date : 2020-02-28
    Andrzej Odrzywolek

    The need for recognition of numerical (decimal, floating-point) constants in terms of elementary functions emerges in many areas of experimental mathematics, numerical analysis, computer algebra systems, model building, approximation and data compression. However, existing solutions are plagued by lack of any criteria distinguishing between random formula, matching literally decimal expansion (i.e

  • A Linear Algebra Approach for Detecting Binomiality of Steady State Ideals of Reversible Chemical Reaction Networks
    arXiv.cs.SC Pub Date : 2020-02-28
    Hamid Rahkooy; Thomas Sturm

    Motivated by problems from Chemical Reaction Network Theory, we investigate whether steady state ideals of reversible reaction networks are generated by binomials. We take an algebraic approach considering, besides concentrations of species, also rate constants as indeterminates. This allows us to represent the generators of a steady state ideal as sums of binomials, which yields a corresponding coefficient

  • Sparse Interpolation With Errors in Chebyshev Basis Beyond Redundant-Block Decoding
    arXiv.cs.SC Pub Date : 2019-12-12
    Erich L. Kaltofen; Zhi-Hong Yang

    We present sparse interpolation algorithms for recovering a polynomial with $\le B$ terms from $N$ evaluations at distinct values for the variable when $\le E$ of the evaluations can be erroneous. Our algorithms perform exact arithmetic in the field of scalars $\mathsf{K}$ and the terms can be standard powers of the variable or Chebyshev polynomials, in which case the characteristic of $\mathsf{K}$

  • Space Efficient Representations of Finite Groups
    arXiv.cs.SC Pub Date : 2020-02-26
    Bireswar Das; Shivdutt Sharma; P. R. Vaidyanathan

    The Cayley table representation of a group uses $\mathcal{O}(n^2)$ words for a group of order $n$ and answers multiplication queries in time $\mathcal{O}(1)$. It is interesting to ask if there is a $o(n^2)$ space representation of groups that still has $\mathcal{O}(1)$ query-time. We show that for any $\delta$, $\frac{1}{\log n} \le \delta \le 1$, there is an $\mathcal{O}(\frac{n^{1 +\delta}}{\delta})$

  • Robust Numerical Tracking of One Path of a Polynomial Homotopy on Parallel Shared Memory Computers
    arXiv.cs.SC Pub Date : 2020-02-21
    Simon Telen; Marc Van Barel; Jan Verschelde

    We consider the problem of tracking one solution path defined by a polynomial homotopy on a parallel shared memory computer. Our robust path tracker applies Newton's method on power series to locate the closest singular parameter value. On top of that, it computes singular values of the Hessians of the polynomials in the homotopy to estimate the distance to the nearest different path. Together, these

  • A Modular Termination Method for Second-Order Computation
    arXiv.cs.SC Pub Date : 2019-12-07
    Makoto Hamana

    We present a new modular proof method of termination for second-order computation, and report its implementation SOL. The proof method is useful for proving termination of higher-order foundational calculi. To establish the method, we use a variation of semantic labelling translation and Blanqui's General Schema: a syntactic criterion of strong normalisation. As an application, we show termination

  • On the Uniqueness of Simultaneous Rational Function Reconstruction
    arXiv.cs.SC Pub Date : 2020-02-20
    Eleonora Guerrini; Romain Lebreton; Ilaria Zappatore

    This paper focuses on the problem of reconstructing a vector of rational functions given some evaluations, or more generally given their remainders modulo different polynomials. The special case of rational functions sharing the same denominator, a.k.a.Simultaneous Rational Function Reconstruction (SRFR), has many applications from linear system solving to coding theory, provided that SRFR has a unique

  • Kleene stars of the plane, polylogarithms and symmetries
    arXiv.cs.SC Pub Date : 2018-11-22
    Gérard Henry Edmond DuchampLIPN; Vincel Hoang Ngoc Minh; Ngo Quoc Hoan

    We extend the definition and construct several bases for polylogarithms Li T , where T are some series, recognizable by a finite state (multiplicity) automaton of alphabet 4 X = {x 0 , x 1 }. The kernel of this new "polylogarithmic map" Li $\bullet$ is also characterized and provides a rewriting process which terminates to a normal form. We concentrate on algebraic and analytic aspects of this extension

  • Probabilistic Condition Number Estimates for Real Polynomial Systems II: Structure and Smoothed Analysis
    arXiv.cs.SC Pub Date : 2018-09-10
    Alperen A. Ergür; Grigoris Paouris; J. Maurice Rojas

    We consider the sensitivity of real zeros of structured polynomial systems to perturbations of their coefficients. In particular, we provide explicit estimates for condition numbers of structured random real polynomial systems, and extend these estimates to smoothed analysis setting.

  • A divide-and-conquer algorithm for computing Gröbner bases of syzygies in finite dimension
    arXiv.cs.SC Pub Date : 2020-02-15
    Simone Naldi; Vincent Neiger

    Let $f_1,\ldots,f_m$ be elements in a quotient $R^n / N$ which has finite dimension as a $K$-vector space, where $R = K[X_1,\ldots,X_r]$ and $N$ is an $R$-submodule of $R^n$. We address the problem of computing a Gr\"obner basis of the module of syzygies of $(f_1,\ldots,f_m)$, that is, of vectors $(p_1,\ldots,p_m) \in R^m$ such that $p_1 f_1 + \cdots + p_m f_m = 0$. An iterative algorithm for this

  • On the Existence of Telescopers for Rational Functions in Three Variables
    arXiv.cs.SC Pub Date : 2019-01-27
    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. For a given class of inputs, when telescopers exist and how to construct telescopers efficiently if they exist are two fundamental problems related

  • ENIGMA Anonymous: Symbol-Independent Inference Guiding Machine (system description)
    arXiv.cs.SC Pub Date : 2020-02-13
    Jan Jakubův; Karel Chvalovský; Miroslav Olšák; Bartosz Piotrowski; Martin Suda; Josef Urban

    We describe an implementation of gradient boosting and neural guidance of saturation-style automated theorem provers that does not depend on consistent symbol names across problems. For the gradient-boosting guidance, we manually create abstracted features by considering arity-based encodings of formulas. For the neural guidance, we use symbol-independent graph neural networks and their embedding of

  • Smooth Points on Semi-algebraic Sets
    arXiv.cs.SC Pub Date : 2020-02-11
    Katherine Harris; Jonathan D. Hauenstein; Agnes Szanto

    Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon the ability to compute smooth points. Existing methods to compute smooth points on semi-algebraic sets use symbolic quantifier elimination tools. In this paper, we present a simple algorithm based on computing the critical points of some well-chosen function that guarantees the computation of smooth points

  • Improving Graph Neural Network Representations of Logical Formulae with Subgraph Pooling
    arXiv.cs.SC Pub Date : 2019-11-15
    Maxwell Crouse; Ibrahim Abdelaziz; Cristina Cornelio; Veronika Thost; Lingfei Wu; Kenneth Forbus; Achille Fokoue

    Recent advances in the integration of deep learning with automated theorem proving have centered around the representation of logical formulae as inputs to deep learning systems. In particular, there has been a growing interest in adapting structure-aware neural methods to work with the underlying graph representations of logical expressions. While more effective than character and token-level approaches

  • Signature-based algorithms for Gr{ö}bner bases over Tate algebras
    arXiv.cs.SC Pub Date : 2020-02-11
    Xavier CarusoLAGA; Tristan VacconXLIM-MATHIS; Thibaut Verron

    Introduced by Tate in [Ta71], Tate algebras play a major role in the context of analytic geometry over the-adics, where they act as a counterpart to the use of polynomial algebras in classical algebraic geometry. In [CVV19] the formalism of Gr{\"o}bner bases over Tate algebras has been introduced and effectively implemented. One of the bottleneck in the algorithms was the time spent on reduction ,

  • SPECTRA -- a Maple library for solving linear matrix inequalities in exact arithmetic
    arXiv.cs.SC Pub Date : 2016-11-07
    Mohab Safey El DinPolSys; Didier HenrionLAAS-MAC, CTU; Simone NaldiTU; Mohab SafeyPolSys; El DinPolSys

    This document describes our freely distributed Maple library {\sc spectra}, for Semidefinite Programming solved Exactly with Computational Tools of Real Algebra. It solves linear matrix inequalities with symbolic computation in exact arithmetic and it is targeted to small-size, possibly degenerate problems for which symbolic infeasibility or feasibility certificates are required.

  • The Fundamental Theorem of Tropical Partial Differential Algebraic Geometry
    arXiv.cs.SC Pub Date : 2020-02-07
    Sebastian Falkensteiner; Cristhian Garay-López; Mercedes Haiech; Marc Paul Noordman; Zeinab Toghani; François Boulier

    Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of Tropical Differential Algebraic Geometry states that the support of solutions of systems of ordinary differential equations with formal power series coefficients over

  • First-Order Tests for Toricity
    arXiv.cs.SC Pub Date : 2020-02-10
    Hamid Rahkooy; Thomas Sturm

    Motivated by problems arising with the symbolic analysis of steady state ideals in Chemical Reaction Network Theory, we consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a coset of a multiplicative group. That property corresponds to Shifted Toricity, a recent generalization of toricity of the corresponding polynomial ideal. The key idea

  • Compatible rewriting of noncommutative polynomials for proving operator identities
    arXiv.cs.SC Pub Date : 2020-02-10
    Cyrille Chenavier; Clemens Hofstadler; Clemens G. Raab; Georg Regensburger

    The goal of this paper is to prove operator identities using equalities between noncommutative polynomials. In general, a polynomial expression is not valid in terms of operators, since it may not be compatible with domains and codomains of the corresponding operators. Recently, some of the authors introduced a framework based on labelled quivers to rigorously translate polynomial identities to operator

  • Sparse Polynomial Interpolation Based on Diversification
    arXiv.cs.SC Pub Date : 2020-01-21
    Qiao-Long Huang

    We consider the problem of interpolating a sparse multivariate polynomial over a finite field, represented with a black box. Building on the algorithm of Ben-Or and Tiwari for interpolating polynomials over rings with characteristic zero, we develop a new Monte Carlo algorithm over the finite field by doing additional probes. To interpolate a polynomial $f\in F_q[x_1,\dots,x_n]$ with a partial degree

  • Sparse Polynomial Interpolation Based on Derivative
    arXiv.cs.SC Pub Date : 2020-01-21
    Qiao-Long Huang

    In this paper, we propose two new interpolation algorithms for sparse multivariate polynomials represented by a straight-line program(SLP). Both of our algorithms work over any finite fields $F_q$ with large characteristic. The first one is a Monte Carlo randomized algorithm. Its arithmetic complexity is linear in the number $T$ of non-zero terms of $f$, in the number $n$ of variables. If $q$ is $O((nTD)^{(1)})$

  • Integral P-Recursive Sequences
    arXiv.cs.SC Pub Date : 2020-02-07
    Shaoshi Chen; Lixin Du; Manuel Kauers; Thibaut Verron

    In an earlier paper, the notion of integrality known from algebraic number fields and fields of algebraic functions has been extended to D-finite functions. The aim of the present paper is to extend the notion to the case of P-recursive sequences. In order to do so, we formulate a general algorithm for finding all integral elements for valued vector spaces and then show that this algorithm includes

  • An Additive Decomposition in S-Primitive Towers
    arXiv.cs.SC Pub Date : 2020-02-06
    Hao Du; Jing Guo; Ziming Li; Elaine Wong

    We consider the additive decomposition problem in primitive towers and present an algorithm to decompose a function in an S-primitive tower as a sum of a derivative in the tower and a remainder which is minimal in some sense. Special instances of S-primitive towers include differential fields generated by finitely many logarithmic functions and logarithmic integrals. A function in an S-primitive tower

  • Convergence analysis of particle swarm optimization using stochastic Lyapunov functions and quantifier elimination
    arXiv.cs.SC Pub Date : 2020-02-05
    Maximilian Gerwien; Rick Voßwinkel; Hendrik Richter

    This paper adds to the discussion about theoretical aspects of particle swarm stability by proposing to employ stochastic Lyapunov functions and to determine the convergence set by quantifier elimination. We present a computational procedure and show that this approach leads to reevaluation and extension of previously know stability regions for PSO using a Lyapunov approach under stagnation assumptions

  • Separating Variables in Bivariate Polynomial Ideals
    arXiv.cs.SC Pub Date : 2020-02-04
    Manfred Buchacher; Manuel Kauers; Gleb Pogudin

    We present an algorithm which for any given ideal $I\subseteq\mathbb{K} [x,y]$ finds all elements of $I$ that have the form $f(x) - g(y)$, i.e., all elements in which no monomial is a multiple of $xy$.

  • Tropical recurrent sequences
    arXiv.cs.SC Pub Date : 2018-07-27
    Dima Grigoriev

    Tropical recurrent sequences are introduced satisfying a given vector (being a tropical counterpart of classical linear recurrent sequences). We consider the case when Newton polygon of the vector has a single (bounded) edge. In this case there are periodic tropical recurrent sequences which are similar to classical linear recurrent sequences. A question is studied when there exists a non-periodic

  • Nearly Optimal Sparse Polynomial Multiplication
    arXiv.cs.SC Pub Date : 2019-01-27
    Vasileios Nakos

    In the sparse polynomial multiplication problem, one is asked to multiply two sparse polynomials f and g in time that is proportional to the size of the input plus the size of the output. The polynomials are given via lists of their coefficients F and G, respectively. Cole and Hariharan (STOC 02) have given a nearly optimal algorithm when the coefficients are positive, and Arnold and Roche (ISSAC 15)

  • Linearly Constrained Gaussian Processes with Boundary Conditions
    arXiv.cs.SC Pub Date : 2020-02-03
    Markus Lange-Hegermann

    One goal in Bayesian machine learning is to encode prior knowledge into prior distributions, to model data efficiently. We consider prior knowledge from systems of linear (partial and ordinary) differential equations together with their boundary conditions. We construct multi-output Gaussian process priors with realizations dense in the solution set of such systems, in particular any solution (and

  • Efficient q-Integer Linear Decomposition of Multivariate Polynomials
    arXiv.cs.SC Pub Date : 2020-02-01
    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 in the q-analogous world of symbolic summation, for example, describing the q-counterpart of Ore-Sato theory or determining the applicability of the q-analogue of Zeilberger's algorithm to a q-hypergeometric term. Both of our algorithms require only

  • Essentially optimal sparse polynomial multiplication
    arXiv.cs.SC Pub Date : 2020-01-31
    Pascal Giorgi; Bruno Grenet; Armelle Perret du Cray

    In this article, we present a probabilistic algorithm to compute the product of two univariate sparse polynomials over a field with a number of bit operations that is quasi-linear in the size of the input and the output. Our algorithm works for any field of characteristic zero or larger than the degree. We mainly rely on sparse interpolation and on a new algorithm for verifying a sparse product that

  • Unsatisfiability Proofs for Weight 16 Codewords in Lam's Problem
    arXiv.cs.SC Pub Date : 2020-01-31
    Curtis Bright; Kevin K. H. Cheung; Brett Stevens; Ilias Kotsireas; Vijay Ganesh

    In the 1970s and 1980s, searches performed by L. Carter, C. Lam, L. Thiel, and S. Swiercz showed that projective planes of order ten with weight 16 codewords do not exist. These searches required highly specialized and optimized computer programs and required about 2,000 hours of computing time on mainframe and supermini computers. In 2011, these searches were verified by D. Roy using an optimized

  • Nonexistence Certificates for Ovals in a Projective Plane of Order Ten
    arXiv.cs.SC Pub Date : 2020-01-31
    Curtis Bright; Kevin K. H. Cheung; Brett Stevens; Ilias Kotsireas; Vijay Ganesh

    In 1983, a computer search was performed for ovals in a projective plane of order ten. The search was exhaustive and negative, implying that such ovals do not exist. However, no nonexistence certificates were produced by this search, and to the best of our knowledge the search has never been independently verified. In this paper, we rerun the search for ovals in a projective plane of order ten and

  • Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials
    arXiv.cs.SC Pub Date : 2020-01-24
    Evelyne Hubert; Michael F. Singer

    Sparse interpolation} refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. Beyond the multivariate Chebyshev polynomial obtained as tensor products of univariate Chebyshev polynomials, the theory of root

  • Freeness and invariants of rational plane curves
    arXiv.cs.SC Pub Date : 2018-04-17
    Laurent BuséAROMATH; Alexandru DimcaJAD, AROMATH; Gabriel Sticlaru

    Given a parameterization $\phi$ of a rational plane curve C, we study some invariants of C via $\phi$. We first focus on the characterization of rational cuspidal curves, in particular we establish a relation between the discriminant of the pull-back of a line via $\phi$, the dual curve of C and its singular points. Then, by analyzing the pull-backs of the global differential forms via $\phi$, we prove

  • Bisimilar Conversion of Multi-valued Networks to Boolean Networks
    arXiv.cs.SC Pub Date : 2020-01-21
    Franck DelaplaceIBISC; Sergiu IvanovLACL

    Discrete modelling frameworks of Biological networks can be divided in two distinct categories: Boolean and Multi-valued. Although Multi-valued networks are more expressive for qualifying the regulatory behaviours modelled by more than two values, the ability to automatically convert them to Boolean network with an equivalent behaviour breaks down the fundamental borders between the two approaches

  • On mu-Symmetric Polynomials
    arXiv.cs.SC Pub Date : 2020-01-21
    Jing Yang; Chee K. Yap

    In this paper, we study functions of the roots of a univariate polynomial in which the roots have a given multiplicity structure $\mu$. Traditionally, root functions are studied via the theory of symmetric polynomials; we extend this theory to $\mu$-symmetric polynomials. We were motivated by a conjecture from Becker et al.~(ISSAC 2016) about the $\mu$-symmetry of a particular root function $D^+(\mu)$

  • On the k-synchronizability of systems
    arXiv.cs.SC Pub Date : 2019-09-04
    Cinzia Di GiustoC&A; Cinzia GiustoSARDES; Laetitia LaversaC&A; Etienne Lozes

    In this paper, we work on the notion of k-synchronizability: a system is k-synchronizable if any of its executions, up to reordering causally independent actions, can be divided into a succession of k-bounded interaction phases. We show two results (both for mailbox and peer-to-peer automata): first, the reachability problem is decidable for k-synchronizable systems; second, the membership problem

  • Effective Coefficient Asymptotics of Multivariate Rational Functions via Semi-Numerical Algorithms for Polynomial Systems
    arXiv.cs.SC Pub Date : 2019-05-10
    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

  • $\mathtt{bimEX}$: A Mathematica package for exact computations in $3+1$ bimetric relativity
    arXiv.cs.SC Pub Date : 2019-04-23
    Francesco Torsello

    We present $\mathtt{bimEX}$, a Mathematica package for exact computations in 3$+$1 bimetric relativity. It is based on the $\mathtt{xAct}$ bundle, which can handle computations involving both abstract tensors and their components. In this communication, we refer to the latter case as concrete computations. The package consists of two main parts. The first part involves the abstract tensors, and focuses

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