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Comparing machine learning models to choose the variable ordering for cylindrical algebraic decompositionApr 24 2019There has been recent interest in the use of machine learning (ML) approaches within mathematical software to make choices that impact on the computing performance without affecting the mathematical correctness of the result. We address the problem of ... More
$\mathtt{bimEX}$: A Mathematica package for exact computations in 3$+$1 bimetric relativityApr 23 2019We 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, ... More
Improved algorithms for left factorial residuesApr 19 2019We present improved algorithms for computing the left factorial residues $!p=0!+1!+...+(p-1)! \!\mod p$. We use these algorithms for the calculation of the residues $!p\!\mod p$, for all primes $p$ up to $2^{40}$. Our results confirm that Kurepa's left ... More
Unification and combination of iterative insertion strategies with one-step traversalsApr 16 2019Motivated by an ongoing project on the computer aided derivation of multiscale partial differential equation models, we introduce a class of term transformations that consists in navigation strategies and insertion of contexts. We define a unification ... More
Asymptotic Solutions of Polynomial Equations with Exp-Log CoefficientsApr 15 2019We present an algorithm for computing asymptotic approximations of roots of polynomials with exp-log function coefficients. The real and imaginary parts of the approximations are given as explicit exp-log expressions. We provide a method for deciding ... More
Proceedings Joint International Workshop on Linearity & Trends in Linear Logic and ApplicationsApr 12 2019This volume contains a selection of papers presented at Linearity/TLLA 2018: Joint Linearity and TLLA workshops (part of FLOC 2018) held on July 7-8, 2018 in Oxford. Linearity has been a key feature in several lines of research in both theoretical and ... More
Tropical Differential Groebner BasisApr 03 2019In this paper, the tropical differential Gr\"obner basis is studied, which is a natural generalization of the tropical Gr\"obner basis to the recently introduced tropical differential algebra. Like the differential Gr\"obner basis, the tropical differential ... More
Exact Lower Bounds for Monochromatic Schur Triples and GeneralizationsApr 03 2019We derive exact and sharp lower bounds for the number of monochromatic generalized Schur triples $(x,y,x+ay)$ whose entries are from the set $\{1,\dots,n\}$, subject to a coloring with two different colors. Previously, only asymptotic formulas for such ... More
Reconstructing Rational Functions with $\texttt{FireFly}$Mar 29 2019We present the open-source $\texttt{C++}$ library $\texttt{FireFly}$ for the reconstruction of multivariate rational functions over finite fields. We discuss the involved algorithms and their implementation. As an application, we use $\texttt{FireFly}$ ... More
Computing huge Groebner basis like cyclic10 over $\Q$ with GiacMar 29 2019We present a short description on how to fine-tune the modular algorithm implemented in the Giac computer algebra system to reconstruct huge Groebner basis over $\Q$.The classical cyclic10 benchmark will serve as example.
Testing zero-dimensionality of varieties at a pointMar 29 2019Effective methods are introduced for testing zero-dimensionality of varieties at a point. The motivation of this paper is to compute and analyze deformations of isolated hypersurface singularities. As an application, methods for computing local dimensions ... More
Local Search for Fast Matrix MultiplicationMar 27 2019Laderman discovered a scheme for computing the product of two 3x3 matrices using only 23 multiplications in 1976. Since then, some more such schemes were proposed, but it remains open how many there are and whether there exist schemes with fewer than ... More
Computations of the Expected Euler Characteristic for the Largest Eigenvalue of a Real Wishart MatrixMar 25 2019We give an approximate formula of the distribution of the largest eigenvalue of real Wishart matrices by the expected Euler characteristic method for the general dimension. The formula is expressed in terms of a definite integral with parameters. We derive ... More
Cylindrical Algebraic Decomposition with Equational ConstraintsMar 20 2019Cylindrical Algebraic Decomposition (CAD) has long been one of the most important algorithms within Symbolic Computation, as a tool to perform quantifier elimination in first order logic over the reals. More recently it is finding prominence in the Satisfiability ... More
On some classes of irreducible polynomialsMar 20 2019The aim of the paper is to produce new families of irreducible polynomials, generalizing previous results in the area. One example of our general result is that for a near-separated polynomial, i.e., polynomials of the form $F(x,y)=f_1(x)f_2(y)-f_2(x)f_1(y)$, ... More
Minimizing polynomial functions on quantum computersMar 19 2019This expository paper reviews some of the recent uses of computational algebraic geometry in classical and quantum optimization. The paper assumes an elementary background in algebraic geometry and adiabatic quantum computing (AQC), and concentrates on ... More
Recursive Matrix Algorithms in Commutative Domain for Cluster with Distributed MemoryMar 11 2019We give an overview of the theoretical results for matrix block-recursive algorithms in commutative domains and present the results of experiments that we conducted with new parallel programs based on these algorithms on a supercomputer MVS-10P at the ... More
Quadratic Probabilistic Algorithms for Normal BasesMar 08 2019It is well known that for any finite Galois extension field $K/F$, with Galois group $G = \mathrm{Gal}(K/F)$, there exists an element $\alpha \in K$ whose orbit $G\cdot\alpha$ forms an $F$-basis of $K$. Such an element $\alpha$ is called \emph{normal} ... More
Discovering and Proving Infinite Pochhammer Sum IdentitiesFeb 28 2019We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi,$ $\log(2)$ or zeta values. In order to perform these simplifications, we view the series as specializations ... More
Discovering and Proving Infinite Pochhammer Sum IdentitiesFeb 28 2019Apr 10 2019We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi,$ $\log(2)$ or zeta values. In order to perform these simplifications, we view the series as specializations ... More
Counting basic-irreducible factors mod $p^k$ in deterministic poly-time and $p$-adic applicationsFeb 20 2019Finding an irreducible factor, of a polynomial $f(x)$ modulo a prime $p$, is not known to be in deterministic polynomial time. Though there is such a classical algorithm that {\em counts} the number of irreducible factors of $f\bmod p$. We can ask the ... More
Computing Minimal Presentations and Betti Numbers of 2-Parameter Persistent HomologyFeb 15 2019Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm assumes that $M$ is given implicitly: It takes as input ... More
Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent HomologyFeb 15 2019Mar 25 2019Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm takes as input a short chain complex of free modules \[ ... More
Identifying the Parametric Occurrence of Multiple Steady States for some Biological NetworksFeb 13 2019We consider a problem from biological network analysis of determining regions in a parameter space over which there are multiple steady states for positive real values of variables and parameters. We describe multiple approaches to address the problem ... More
When Causal Intervention Meets Image Masking and Adversarial Perturbation for Deep Neural NetworksFeb 09 2019Feb 13 2019Discovering and exploiting the causality in deep neural networks (DNNs) are crucial challenges for understanding and reasoning causal effects (CE) on an explainable visual model. "Intervention" has been widely used for recognizing a causal relation ontologically. ... More
When Causal Intervention Meets Image Masking and Adversarial Perturbation for Deep Neural NetworksFeb 09 2019Discovering and exploiting the causality in deep neural networks (DNNs) are crucial challenges for understanding and reasoning causal effects (CE) on an explainable visual model. "Intervention" has been widely used for recognizing a causal relation ontologically. ... More
Generic reductions for in-place polynomial multiplicationFeb 08 2019The polynomial multiplication problem has attracted considerable attention since the early days of computer algebra, and several algorithms have been designed to achieve the best possible time complexity. More recently, efforts have been made to improve ... More
Exact Optimization via Sums of Nonnegative Circuits and Sums of AM/GM ExponentialsFeb 06 2019We provide two hybrid numeric-symbolic optimization algorithms, computing exact sums of nonnegative circuits (SONC) and sums of arithmetic-geometric-exponentials (SAGE) decompositions. Moreover, we provide a hybrid numeric-symbolic decision algorithm ... More
Modeling Terms by Graphs with Structure Constraints (Two Illustrations)Feb 06 2019In the talk at the workshop my aim was to demonstrate the usefulness of graph techniques for tackling problems that have been studied predominantly as problems on the term level: increasing sharing in functional programs, and addressing questions about ... More
On the Complexity of Toric IdealsFeb 04 2019We investigate the computational complexity of problems on toric ideals such as normal forms, Gr\"obner bases, and Graver bases. We show that all these problems are strongly NP-hard in the general case. Nonetheless, we can derive efficient algorithms ... More
Gr{ö}bner Basis over Semigroup Algebras: Algorithms and Applications for Sparse Polynomial SystemsFeb 01 2019Gr{\"o}bner bases is one the most powerful tools in algorithmic non-linear algebra. Their computation is an intrinsically hard problem with a complexity at least single exponential in the number of variables. However, in most of the cases, the polynomial ... More
Algorithmic counting of nonequivalent compact Huffman codesJan 31 2019It is known that the following five counting problems lead to the same integer sequence~$f_t(n)$: the number of nonequivalent compact Huffman codes of length~$n$ over an alphabet of $t$ letters, the number of `nonequivalent' canonical rooted $t$-ary trees ... More
Rational Solutions of First-Order Algebraic Ordinary Difference EquationsJan 30 2019Feb 01 2019We propose an algebraic geometric approach for studying rational solutions of first-order algebraic ordinary difference equations. For an autonomous first-order algebraic ordinary difference equations, we give an upper bound for the degrees of its rational ... More
LU factorization with errors *Jan 30 2019We present new algorithms to detect and correct errors in the lower-upper factorization of a matrix, or the triangular linear system solution, over an arbitrary field. Our main algorithms do not require any additional information or encoding other than ... More
An Optimization-Based Sum-of-Squares Approach to Vizing's ConjectureJan 29 2019Feb 04 2019Vizing's conjecture (open since 1968) relates the sizes of dominating sets in two graphs to the size of a dominating set in their Cartesian product graph. In this paper, we formulate Vizing's conjecture itself as a Positivstellensatz existence question. ... More
A Faster Solution to Smale's 17th Problem I: Real Binomial SystemsJan 28 2019Suppose $F:=(f_1,\ldots,f_n)$ is a system of random $n$-variate polynomials with $f_i$ having degree $\leq\!d_i$ and the coefficient of $x^{a_1}_1\cdots x^{a_n}_n$ in $f_i$ being an independent complex Gaussian of mean $0$ and variance $\frac{d_i!}{a_1!\cdots ... More
On the Complexity of Computing the Topology of Real Algebraic Space CurvesJan 28 2019In this paper, we present a deterministic algorithm to find a strong generic position for an algebraic space curve. We modify our existing algorithm for computing the topology of an algebraic space curve and analyze the bit complexity of the algorithm. ... More
Signature-based Möller's algorithm for strong Gröbner bases over PIDsJan 28 2019Signature-based algorithms are the latest and most efficient approach as of today to compute Gr\"obner bases for polynomial systems over fields. Recently, possible extensions of these techniques to general rings have attracted the attention of several ... More
Gr{ö}bner bases over Tate algebrasJan 28 2019Tate algebras are fundamental objects in the context of analytic geometry over the p-adics. Roughly speaking, they play the same role as polynomial algebras play in classical algebraic geometry. In the present article, we develop the formalism of Gr{\"o}bner ... More
Existence Problem of Telescopers for Rational Functions in Three Variables: the Mixed CasesJan 27 2019We present criteria on the existence of telescopers for trivariate rational functions in four mixed cases, in which discrete and continuous variables appear simultaneously. We reduce the existence problem in the trivariate case to the exactness testing ... More
Nearly Optimal Sparse Polynomial MultiplicationJan 27 2019In the sparse 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 $G$, ... More
Symbolic integration of hyperexponential 1-formsJan 25 2019Let $H$ be a hyperexponential function in $n$ variables $x=(x_1,\dots,x_n)$ with coefficients in a field $\mathbb{K}$, $[\mathbb{K}:\mathbb{Q}] <\infty$, and $\omega$ a rational differential $1$-form. Assume that $H\omega$ is closed and $H$ transcendental. ... More
Effective certification of approximate solutions to systems of equations involving analytic functionsJan 24 2019We develop algorithms for certifying an approximation to a nonsingular solution of a square system of equations built from univariate analytic functions. These algorithms are based on the existence of oracles for evaluating basic data about the input ... More
Efficiently factoring polynomials modulo $p^4$Jan 20 2019Polynomial factoring has famous practical algorithms over fields-- finite, rational \& $p$-adic. However, modulo prime powers it gets hard as there is non-unique factorization and a combinatorial blowup ensues. For example, $x^2+p \bmod p^2$ is irreducible, ... More
Automated Synthesis of Safe Digital Controllers for Sampled-Data Stochastic Nonlinear SystemsJan 10 2019We present a new method for the automated synthesis of digital controllers with formal safety guarantees for systems with nonlinear dynamics, noisy output measurements, and stochastic disturbances. Our method derives digital controllers such that the ... More
Spectral Approach to Verifying Non-linear Arithmetic CircuitsJan 09 2019This paper presents a fast and effective computer algebraic method for analyzing and verifying non-linear integer arithmetic circuits using a novel algebraic spectral model. It introduces a concept of algebraic spectrum, a numerical form of polynomial ... More
On Fast Matrix Inversion via Fast Matrix MultiplicationJan 03 2019Volker Strassen first suggested an algorithm to multiply matrices with worst case running time less than the conventional $\mathcal{O}(n^3)$ operations in 1969. He also presented a recursive algorithm with which to invert matrices, and calculate determinants ... More
Subresultants of $(x-α)^m$ and $(x-β)^n$, Jacobi polynomials and complexityDec 31 2018In an earlier article together with Carlos D'Andrea [BDKSV2017], we described explicit expressions for the coefficients of the order-$d$ polynomial subresultant of $(x-\alpha)^m$ and $(x-\beta)^n $ with respect to Bernstein's set of polynomials $\{(x-\alpha)^j(x-\beta)^{d-j}, ... More
A New Deflation Method For Verifying the Isolated Singular Zeros of Polynomial SystemsDec 30 2018In this paper, we develop a new deflation technique for refining or verifying the isolated singular zeros of polynomial systems. Starting from a polynomial system with an isolated singular zero, by computing the derivatives of the input polynomials directly ... More
Computing the topology of a planar or space hyperelliptic curveDec 30 2018We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose topology is ... More
SIAN: software for structural identifiability analysis of ODE modelsDec 26 2018Biological processes are often modeled by ordinary differential equations with unknown parameters. The unknown parameters are usually estimated from experimental data. In some cases, due to the structure of the model, this estimation problem does not ... More
A new algorithm for irreducible decomposition of representations of finite groupsDec 23 2018An algorithm for irreducible decomposition of representations of finite groups over fields of characteristic zero is described. The algorithm uses the fact that the decomposition induces a partition of the invariant inner product into a complete set of ... More
The Complexity of Factors of Multivariate PolynomialsDec 17 2018The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such families of ... More
In Praise of Sequence (Co-)Algebra and its implementation in HaskellDec 14 2018Feb 28 2019What is Sequence Algebra? This is a question that any teacher or student of mathematics or computer science can engage with. Sequences are in Calculus, Combinatorics, Statistics and Computation. They are foundational, a step up from number arithmetic. ... More
In Praise of Sequence (Co-)Algebra and its implementation in HaskellDec 14 2018What is Sequence Algebra? This is a question that any teacher or student of mathematics or computer science can engage with. Sequences are in Calculus, Combinatorics, Statistics and Computation. They are foundational, a step up from number arithmetic. ... More
Computing Nearby Non-trivial Smith FormsDec 11 2018We consider the problem of computing the nearest matrix polynomial with a non-trivial Smith Normal Form. We show that computing the Smith form of a matrix polynomial is amenable to numeric computation as an optimization problem. Furthermore, we describe ... More
BOSPHORUS: Bridging ANF and CNF SolversDec 11 2018Algebraic Normal Form (ANF) and Conjunctive Normal Form (CNF) are commonly used to encode problems in Boolean algebra. ANFs are typically solved via Gr"obner basis algorithms, often using more memory than is feasible; while CNFs are solved using SAT solvers, ... More
An efficient reduction strategy for signature-based algorithms to compute Groebner basisNov 30 2018This paper introduces a strategy for signature-based algorithms to compute Groebner basis. The signature-based algorithms generate S-pairs instead of S-polynomials, and use s-reduction instead of the usual reduction used in the Buchberger algorithm. There ... More
On Exact Polya, Hilbert-Artin and Putinar's RepresentationsNov 25 2018We 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 ... More
Chordal Graphs in Triangular Decomposition in Top-Down StyleNov 25 2018In this paper, we first prove that when the associated graph of a polynomial set is chordal, a particular triangular set computed by a general algorithm in top-down style for computing the triangular decomposition of this polynomial set has an associated ... More
Fast Algorithms for Computing Eigenvectors of Matrices via Pseudo Annihilating PolynomialsNov 22 2018An efficient algorithm for computing eigenvectors of a matrix of integers by exact computation is proposed. The components of calculated eigenvectors are expressed as polynomials in the eigenvalue to which the eigenvector is associated, as a variable. ... More
Fast Algorithms for Computing Eigenvectors of Matrices via Pseudo Annihilating PolynomialsNov 22 2018Feb 17 2019An efficient algorithm for computing eigenvectors of a matrix of integers by exact computation is proposed. The components of calculated eigenvectors are expressed as polynomials in the eigenvalue to which the eigenvector is associated, as a variable. ... More
Kleene stars of the plane, polylogarithms and symmetriesNov 22 2018We 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 ... More
Linear Differential Equations as a Data-StructureNov 21 2018A lot of information concerning solutions of linear differential equations can be computed directly from the equation. It is therefore natural to consider these equations as a data-structure, from which mathematical properties can be computed. A variety ... More
A Fast Randomized Geometric Algorithm for Computing Riemann-Roch SpacesNov 20 2018Nov 30 2018We propose a probabilistic Las Vegas variant of Brill-Noether's algorithm for computing a basis of the Riemann-Roch space $L(D)$ associated to a divisor $D$ on a projective plane curve $\mathcal C$ over a sufficiently large perfect field $k$. Our main ... More
Temporal viability regulation for control affine systems with applications to mobile vehicle coordination under time-varying motion constraintsNov 15 2018Controlled invariant set and viability regulation of dynamical control systems have played important roles in many control and coordination applications. In this paper we develop a temporal viability regulation theory for general dynamical control systems, ... More
Staging Human-computer Dialogs: An Application of the Futamura ProjectionsNov 13 2018We demonstrate an application of the Futamura Projections to human-computer interaction, and particularly to staging human-computer dialogs. Specifically, by providing staging analogs to the classical Futamura Projections, we demonstrate that the Futamura ... More
An Application of Rubi: Series Expansion of the Quark Mass Renormalization Group EquationNov 12 2018We highlight how Rule-based Integration (Rubi) is an enhanced method of symbolic integration which allows for the integration of many difficult integrals not accomplished by other computer algebra systems. Using Rubi, many integration techniques become ... More
Complexity Estimates for Fourier-Motzkin EliminationNov 05 2018In this paper, we propose a new method for removing all the redundant inequalities generated by Fourier-Motzkin Elimination. This method is based on Kohler's work and an improved version of Balas' work. Moreover, this method only uses arithmetic operations ... More
Putting Fürer Algorithm into Practice with the BPAS LibraryNov 05 2018Fast algorithms for integer and polynomial multiplication play an important role in scientific computing as well as in other disciplines. In 1971, Sch{\"o}nhage and Strassen designed an algorithm that improved the multiplication time for two integers ... More
A nearly optimal algorithm to decompose binary formsOct 30 2018Symmetric tensor decomposition is an important problem with applications in several areas for example signal processing, statistics, data analysis and computational neuroscience. It is equivalent to Waring's problem for homogeneous polynomials, that is ... More
Learning Distributed Representations of Symbolic Structure Using Binding and Unbinding OperationsOct 29 2018Jan 31 2019Widely used recurrent units, including Long-short Term Memory (LSTM) and Gated Recurrent Unit (GRU), perform well on natural language tasks, but their ability to learn structured representations is still questionable. Exploiting Tensor Product Representations ... More
Reducing the complexity for class group computations using small defining polynomialsOct 29 2018In this paper, we describe an algorithm that efficiently collect relations in class groups of number fields defined by a small defining polynomial. This conditional improvement consists in testing directly the smoothness of principal ideals generated ... More
On the complexity of class group computations for large degree number fieldsOct 26 2018In this paper, we examine the general algorithm for class group computations, when we do not have a small defining polynomial for the number field. Based on a result of Biasse and Fieker, we simplify their algorithm, improve the complexity analysis and ... More
Counting points on hyperelliptic curves with explicit real multiplication in arbitrary genusOct 25 2018We present a probabilistic Las Vegas algorithm for computing the local zeta function of a genus-$g$ hyperelliptic curve defined over $\mathbb F_q$ with explicit real multiplication (RM) by an order $\Z[\eta]$ in a degree-$g$ totally real number field. ... More
Computation of gcd chain over the power of an irreducible polynomialOct 22 2018Dec 27 2018A notion of gcd chain has been introduced by the author at ISSAC 2017 for two univariate monic polynomials with coefficients in a ring R = k[x_1, ..., x_n ]/(T) where T is a primary triangular set of dimension zero. A complete algorithm to compute such ... More
Integration in terms of polylogarithmOct 13 2018This paper provides a Liouville principle for integration in terms of dilogarithm and partial result for polylogarithm.
Computing Elimination Ideals and Discriminants of Likelihood EquationsOct 12 2018We develop a probabilistic algorithm for computing elimination ideals of likelihood equations, which is for larger models by far more efficient than directly computing Groebner bases or the interpolation method proposed in the first author's previous ... More
Reconstruction of surfaces with ordinary singularities from their silhouettesOct 12 2018We present algorithms for reconstructing, up to unavoidable projective automorphisms, surfaces with ordinary singularities in three dimensional space starting from their silhouette, or "apparent contour" - namely the branching locus of a projection on ... More
Synthesis for Vesicle Traffic SystemsOct 10 2018Vesicle Traffic Systems (VTSs) are the material transport mechanisms among the compartments inside the biological cells. The compartments are viewed as nodes that are labeled with the containing chemicals and the transport channels are similarly viewed ... More
Proceedings of the 15th International Workshop on the ACL2 Theorem Prover and Its ApplicationsOct 09 2018Oct 29 2018This volume contains the proceedings of the Fifteenth International Workshop on the ACL2 Theorem Prover and Its Applications (ACL2-2018), a two-day workshop held in Austin, Texas, USA, on November 5-6, 2018, immediately after FMCAD'18. The proceedings ... More
A Novel Algebraic Geometry Compiling Framework for Adiabatic Quantum ComputationsOct 02 2018Adiabatic Quantum Computing (AQC) is an attractive paradigm for solving hard integer polynomial optimization problems. Available hardware restricts the Hamiltonians to be of a structure that allows only pairwise interactions. This requires that the original ... More
Computation of Pommaret Bases Using SyzygiesSep 28 2018We investigate the application of syzygies for efficiently computing (finite) Pommaret bases. For this purpose, we first describe a non-trivial variant of Gerdt's algorithm to construct an involutive basis for the input ideal as well as an involutive ... More
A Pommaret Bases Approach to the Degree of a Polynomial IdealSep 28 2018In this paper, we study first the relationship between Pommaret bases and Hilbert series. Given a finite Pommaret basis, we derive new explicit formulas for the Hilbert series and for the degree of the ideal generated by it which exhibit more clearly ... More
Bohemian Upper Hessenberg Toeplitz MatricesSep 27 2018We look at Bohemian matrices, specifically those with entries from $\{-1, 0, {+1}\}$. More, we specialize the matrices to be upper Hessenberg, with subdiagonal entries $1$. Even more, we consider Toeplitz matrices of this kind. Many properties remain ... More
Bohemian Upper Hessenberg MatricesSep 27 2018We look at Bohemian matrices, specifically those with entries from $\{-1, 0, {+1}\}$. More, we specialize the matrices to be upper Hessenberg, with subdiagonal entries $\pm1$. Many properties remain after these specializations, some of which surprised ... More
Machine-Assisted Proofs (ICM 2018 Panel)Sep 21 2018This submission to arXiv is the report of a panel session at the 2018 International Congress of Mathematicians (Rio de Janeiro, August). It is intended that, while v1 is that report, this stays a living document containing the panelists', and others', ... More
Towards a symbolic summation theory for unspecified sequencesSep 18 2018The article addresses the problem whether indefinite double sums involving a generic sequence can be simplified in terms of indefinite single sums. Depending on the structure of the double sum, the proposed summation machinery may provide such a simplification ... More
Degree bound for toric envelope of a linear algebraic groupSep 18 2018Apr 07 2019Algorithms working with linear algebraic groups often represent them via defining polynomial equations. One can always choose defining equations for an algebraic group to be of the degree at most the degree of the group as an algebraic variety. However, ... More
Computer algebra tools for Feynman integrals and related multi-sumsSep 17 2018In perturbative calculations, e.g., in the setting of Quantum Chromodynamics (QCD) one aims at the evaluation of Feynman integrals. Here one is often faced with the problem to simplify multiple nested integrals or sums to expressions in terms of indefinite ... More
Probabilistic Condition Number Estimates for Real Polynomial Systems II: Structure and Smoothed AnalysisSep 10 2018We consider the sensitivity of real zeros of polynomial systems with respect to perturbation of the coefficients, and extend our earlier probabilistic estimates for the condition number in two directions: (1) We give refined bounds for the condition number ... More
Detecting tropical defects of polynomial equationsSep 10 2018We introduce the notion of tropical defects, certificates that a system of polynomial equations is not a tropical basis, and provide algorithms for finding them around affine spaces of complementary dimension to the zero set. We use these techniques to ... More
Randomized Polynomial-Time Root Counting in Prime Power RingsAug 30 2018Feb 14 2019Suppose $k,p\!\in\!\mathbb{N}$ with $p$ prime and $f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial with degree $d$ and all coefficients having absolute value less than $p^k$. We give a Las Vegas randomized algorithm that computes the number of roots ... More
Aligator.jl - A Julia Package for Loop Invariant GenerationAug 16 2018We describe the Aligator.jl software package for automatically generating all polynomial invariants of the rich class of extended P-solvable loops with nested conditionals. Aligator.jl is written in the programming language Julia and is open-source. Aligator.jl ... More
Bringing Together Dynamic Geometry Software and the Graphics Processing UnitAug 14 2018We equip dynamic geometry software (DGS) with a user-friendly method that enables massively parallel calculations on the graphics processing unit (GPU). This interplay of DGS and GPU opens up various applications in education and mathematical research. ... More
An Effective Framework for Constructing Exponent Lattice Basis of Nonzero Algebraic NumbersAug 08 2018Jan 29 2019Computing a basis for the exponent lattice of algebraic numbers is a basic problem in the field of computational number theory with applications to many other areas. The main cost of a well-known algorithm \cite{ge1993algorithms,kauers2005algorithms} ... More
Minimal solutions of the rational interpolation problemAug 07 2018We compute minimal solutions of the rational interpolation problem in terms of different notions of degrees associated to these functions. In all the cases, the rational interpolating functions with smallest degree can be computed via the Extended Euclidean ... More
Tropical recurrent sequencesJul 27 2018Nov 07 2018Tropical 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 ... More
Tropical recurrent sequencesJul 27 2018Mar 22 2019Tropical 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 ... More
Multiparametric qualimetric microsurgical scanning chip-lancet model: theoretical metrological and biomedical considerationsJul 23 2018The construction of a novel surgical instrument is considered, which is also a probing device providing a signal to the measuring equipment, which after its interpretation allows to obtain useful information about the section quality and the biomaterial ... More